StudySmarter - The all-in-one study app.

4.8 • +11k Ratings

More than 3 Million Downloads

Free

Continuity Over an Interval

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Vectors in Space
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Point Estimation
- Probability
- Probability Calculations
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Quartiles
- Random Variables
- Randomized Block Design
- Residuals
- Sample Mean
- Sample Proportion
- Sampling
- Sampling Distribution
- Scatter Graphs
- Single Variable Data
- Standard Deviation
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Survey Bias
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Venn Diagrams

In our article on Continuity, we looked at the criterion that functions need to be continuous at a point. What if we were asked to analyze the continuity of a function over a whole interval instead? One way is to prove that it is continuous at every single point, but that's impossible since there is an infinite set of points in any given interval. In this article, we will look at functions that are continuous over their domain and some theorems related to continuity over an interval.

For the definition of function continuity at a point, see Continuity.

First, let's review a few things about intervals. Remember that intervals can be open or closed, and you can write them in various notations. The most common are interval notation and inequality notation:

Interval Notation | Inequality Notation |

\( [a, b] \) | \( a \le x \le b \) |

\( (a, b) \) | \( a < x < b \) |

\( (-\infty, \infty ) \) | all real numbers, also written as \( \mathbb{R} \) |

\( [a, b) \) | \( a \le x < b \) |

\( (a,b] \) | \( a < x \le b \) |

More details on interval notation can be found in our article Functions

In addition, let's remind ourselves of a few key terms we will be using in this article: Interior points and endpoints.

An endpoint is a point at the left or right end of an interval.

A point \( x \) is in the **interior** of an interval \( I \) if \( x \in I \) and \( x \) is not an endpoint of \( I \).

Let's do a couple of examples with applications to interval notation.

For the interval \( [2, 3) \) what are the endpoints, and what is in the interior?

Answer:

The endpoints are \( x = 2 \) and \( x = 3 \). Notice that one of the endpoints is in the interval , but the other endpoint is not. Endpoints do not need to be inside the interval.

For the interior, you know that it can't be an endpoint, and it has to be inside the interval. So for this example, any point between 2 and 3 is in the interior, or in other words, points where \( 2 < x < 3 \). This is the same as the interval \( (2,3) \)!

What if your interval is \( (-\infty, \infty ) \)? Does this interval have endpoints? What points are in the interior?

Answer:

The endpoints of an interval need to be numbers; in this case, they aren't (infinity isn't a number), so there are no endpoints.

In fact, any real number is between \( -\infty \) and \( \infty \), so any real number is in the interior. That means the interior is \( \mathbb{R} \).

Since there are endpoints and interior points of intervals, the definition of continuity over an interval needs to take both of them into account. But it is a good idea to use things you already know about continuity at a point and limits from the left and right. So let's start by defining continuity from the left and right.

A function \( f(x) \) is said to be **continuous from the right** at \( a \) if

\[ \lim\limits_{x \to a^+} f(x) = f(a) . \]

A function is said to be **continuous from the left** at \( a \) if

\[ \lim\limits_{x \to a^-} f(x) = f(a). \]

For more information on limits from the left and right, see One-Sided Limits.

The problem with defining continuity over an interval is that there are many different kinds of intervals. Sometimes the endpoints are in the interval, and sometimes they are not. So the definition needs to take all of those cases into account! Let's put together a wish list of what should go into the definition:

Remember that for a function to have a hope of being continuous at a point, the function needs to be defined at that point. So the first part is ensuring that the interval you care about is in the function's domain.

Interior points of the interval are easier since we know we can evaluate the limit of the function there. So the definition needs to say that the function is continuous at any interior point of the interval.

You don't know if the interval has a left endpoint that is in the interval or not. In fact, the left endpoint of the interval might not exist as in the example \( ( -\infty, 0] \). So the definition needs to say something like "if the left endpoint is in the interval then the function is continuous from the left there".

You also don't know if the interval has a right endpoint that is in the interval, so the definition needs to take care of that case, similar to how it takes care of the left endpoint.

Condensing the wish list down into math-speak gives the following:

Let \( I \) be an interval in the domain of the function \( f(x)\). We say that \( f(x) \) is **continuous on the interval** \( I \) if all of the following are true:

- \( f(x) \) is continuous at all interior points of \( I \);
- if the left endpoint \( a \) of the interval \( I \) is in the interval, then \( f(x) \) is continuous from the right at \( a \); and
- if the right endpoint \( b \) of the interval \( I \) is in the interval, then \( f(x) \) is continuous from the left at \( b\).

Sometimes you will want to look at the entire domain of a function and say whether or not it is continuous on the whole domain.

A function \( f(x) \) is said to be **continuous on its domain** if it is continuous at every point in its domain.

Sometimes you will see a function which is continuous on the whole real line called **continuous everywhere**.

Now you can write down the steps to checking if a function is continuous on an interval.

Step 1: Make sure the interval you care about is part of the function's domain.

Step 2: Check the interior of the interval to see if the function is continuous there.

Step 3: Check to see if it is continuous from the right or left as needed at the endpoints of the interval.

First, let's look at some examples of using the definition to see whether or not a function is continuous on an interval.

Is the function \( f(x) = \sqrt{ x-4} \) continuous on the interval \( [0, 7) \)?

Answer:

Step 1: The first step is to check the function's domain. You know that you can't take the square root of a negative number and end up with a real number, so for a point to be in the domain you need \( x - 4 \ge 0 \), or in other words \( x \ge 4 \). Writing that in interval notation, the domain of \( f(x) \) is \( [4, \infty ) \). Since part of the interval \( [0, 7 ) \) isn't in the domain, the function is definitely not continuous on the interval \( [0, 7 ) \) . You don't even need to do the other steps because it has already failed to be continuous.

Is the function \( f(x) = \sqrt{ x-4} \) continuous on the interval \( [4, 7) \)?

Answer:

From the previous example, you already know that the interval \( [4, 7) \) is in the domain of the function, so Step 1 is covered. Since 7 is not in the interval, that means you only need to check two things:

Step 2: Is the function continuous at any point in the interior? The interior of the interval is \( (4, 7) \), so in other words, if you pick a random point \( p \in (4, 7) \), is the function continuous there?

Step 3: Is the function continuous from the right at \( p = 4\)?

Step 2: Now, pick a random point \( p \in (4, 7) \) . You know that this point is in the interior of the interval and that the function is defined there. So using properties of limits and square roots, you get

\[ \lim\limits_{x \to p^+} f(x) = \lim\limits_{x \to p^+} \sqrt{ x-4} = \sqrt{ p-4} = f(p). \]

But was just any old point in the interior of the interval, which means this works for any point in the interior.

Step 3: First, look at the function value at \( p = 4\). You get \( f(4) = \sqrt{4-4} = 0 \). Then using properties of limits and the square root function, taking the limit from the right at \( p = 4\) you get

\[ \lim\limits_{x \to 4^+} f(x) = \lim\limits_{x \to 4^+} \sqrt{ x-4} = 0. \]

Since this is equal to the function value, you know the function is continuous from the right at \( p = 4 \) .

Putting all the steps together, you know that \( f(x) \) is continuous on the interval \( [4, 7) \) .

Looking at a graph of a function, one question you can ask is "what are the intervals where it is continuous"?

The intervals where the function is continuous are called the **intervals of continuity**.

For the function in the graph, is it continuous on the interval \( (-2, 3] \)?

Answer:

Notice that the interval does not contain the left endpoint but does contain the right endpoint.

Going through the steps to check for continuity on an interval:

Step 1: The function is defined on the entire interval, so that part is good to go.

Step 2: Now, you need to check points in the interior to make sure the function is continuous there. The interior of \( (-2, 3] \) is the interval \( (-2, 3) \) . If you pick any point in the interior and look at the limit, it is certainly the same as the function value. That means the function is continuous in the interior.

Step 3: So you just need to check that \( f(x) \) is continuous from the left at \( x = 3 \) because it is in the interval. You don't need to check that \( f(x) \) is continuous from the right at \( x = -2 \) because it isn't in the interval. As you can see from the graph,

\[ \lim\limits_{x \to 3^-} f(x) = 7 = f(3), \]

so the function is continuous from the left at \( x = 3 \) .

Hence the function \( f(x) \) is continuous on the interval \( (-2, 3] \) .

For the function in the graph, is it continuous on the interval \( (-2, 3] \) ?

Answer:

This function is almost the same as the one in the previous example. In fact, the check to make sure it is continuous in the interior is exactly the same. So it just remains to check the right endpoint of the interval \( (-2, 3] \) . Notice that \( f(3) = 1 \). Also,

\[ \lim\limits_{x \to 3^-} f(x) = 7 . \]

Now the limit from the left at the endpoint is not the same as the function value, so the function is NOT continuous on the interval \( (-2, 3] \) .

For the function in the graph below, find all intervals of continuity.

Answer:

It is clear looking at the picture that the function is defined everywhere. Even at \( x = 0 \), the function is defined, and \( f(0) = 3 \). In addition, everywhere other than \( x = 0 \) the limit is the same as the function value. So, the only point you need to be concerned about is \( x = 0 \). Since this point is in the domain, you need to check the limit from the left and right:

\[ \lim\limits_{x \to 0^-} f(x) = 3 , \]

and

\[ \lim\limits_{x \to 0^+} f(x) = \infty . \]

Since those two limits are not the same, the function is not continuous at \( x = 0 \) even though it is defined there. So the intervals of continuity are \( (-\infty , 0) \cup ( 0, \infty ) \).

Naturally, you don't want to graph every function to see where the intervals of continuity are. So let's look at some examples of using the formula of the function to find them.

Find the intervals of continuity for the function

\[ f(x) = \frac{x + 3}{\sqrt{ x^2 - 4}} . \]

Answer:

The steps are exactly the same if you are looking for intervals of continuity.

Step 1: To start looking for intervals of continuity, first, you need to find the domain of the function. The numerator of this function is a nice line, and it is defined everywhere.

So the only problem would be the denominator of the function, which is \( \sqrt{ x^2 - 4} \).

Remember that you can't have a zero in the denominator, and you can't take the square root of a negative number. You can factor this equation to get:

\[ \sqrt{ x^2 - 4} = \sqrt{ (x - 2)(x+ 2)} , \]

and the roots of this equation are at \( x = 2 \) and \( x = -2 \). In addition, it is only defined when

\[ x^2 - 4 \ge 0 , \]

or in other words when

\[ x^2 \ge 4 \]

which means either \( x \le -2 \) or \( x \ge 2 \) must be true. Then putting together the "no zero in the denominator" rule with the "positive numbers inside square roots" rule, you can see that the function \( f(x) \) has the domain \( ( -\infty , -2) \cup (2, \infty ) \).

Step 2: There are no other possible points of discontinuity in the domain other than the ones you have already found, so the function is continuous on the interior of the domain.

Step 3: The domain's endpoints aren't in the domain, so you don't need to do a special check for them.

That means the intervals of continuity for \( f(x) \) are \( ( -\infty , -2) \) and \( (2, \infty ) \).

Find the intervals of continuity for the function

\[ f(x) = \sqrt{ -x^3 -3x^2 + 13x + 15} . \]

Answer:

Step 1: The first step is to find the domain of the function. It helps that the function inside the square root has a factored form and that

\[ -x^3 -3x^2 + 13x + 15 = -(x-3)(x+1)(x+5). \]

The roots of this function are at \( x = 3 \), \( x = -1 \), and \( x = -5 \).

Plugging in test values at each interval between the roots tells us that the function

\[ y = -(x-3)(x+1)(x+5) \]

is positive on the intervals \( (-\infty , -5) \cup (-1, 3) \). So the domain of \( f(x) \) is \( (-\infty , -5] \cup [-1, 3] \) .

More details on how to determine if the function is positive or negative on an interval can be found in the Interval Notation section of our article on Functions

Step 2: The interior of the domain is \( (-\infty , -5) \cup (-1, 3) \) , and there are no possible points of discontinuity there. That means the function is continuous on the interior of the domain.

Step 3: You just need to check that the left or right limits at the domain endpoints are the same as the function values there.

For \( x = -5 \), evaluating gives \( f(-5) = 0 \). Looking at the limit from the left there,

\[ \lim\limits_{x \to -5^-} f(x) = \lim\limits_{x \to -5^-} \sqrt{ -x^3 -3x^2 + 13x + 15} = 0 . \]

Since the two are the same, \( f(x) \) is continuous from the left at \( x = -5 \) . We can do a similar check to show that is continuous from the left at \( x = 3 \).

For \( x = -1 \) you will need to check the limit from the right. So

\[ \lim\limits_{x \to -1^+} f(x) = \lim\limits_{x \to -1^+} \sqrt{ -x^3 -3x^2 + 13x + 15} = 0 = f(-1), \]

which means that is continuous from the right at \( x = -1 \) .

Putting it all together, you have checked the interior of the domain, and the appropriate limits from the left and right at the endpoints of the domain that are actually in the domain, so you know that \( f(x) \) is continuous on its domain. That means the intervals of continuity are \( (-\infty , -5) \) and \( (-1, 3) \).

Let's apply the definition of continuity over an interval to some examples.

Take the line \( f(x) = 4x - 3 \). The domain of this function is the whole real line. Is this function continuous everywhere?

Answer:

Rather than trying to do this for every single point in the domain (which would be impossible!) take \( p \) to be a random real number. Using the definition of continuous, you need to check that the limit exists at \( p \) and is the same as the function value there. Checking it gives

\[ \lim\limits_{x \to p} f(x) = \lim\limits_{x \to p} (4x-3) = 4p-3 = f(p), \]

which means that \( f(x) \) is continuous at \( p \). But \( p \) was just a random real number which means this works for any real number! Therefore \( f(x) \) is continuous everywhere.

In fact, you can do exactly the same process as in the previous example for any polynomial, leading to the following theorem.

**Theorem:** Every polynomial is continuous on the whole real line.

What about rational functions?

Where is the function

\[ f(x) = \frac{ x^2 - 4}{x + 3} \]

continuous?

Answer:

First, you need to decide where the domain of the function is because you certainly don't want to waste time checking points that aren't in the domain. You already know that the domain of rational functions is everywhere except where the denominator is equal to zero. That means the domain of is \( ( -\infty, -3) \cup (-3, \infty ) \). Just like in the previous example, take \( p \) to be a random point in the domain, so \( p \in ( -\infty, -3) \cup (-3, \infty ) \) . Because \( p \) is in the domain, you know that \( p \) isn't the endpoint of the domain (in other words, it isn't \( -3 \) ), so you don't need to check left or right limits. Checking the limit,

\[ \lim\limits_{x \to p} f(x) = \lim\limits_{x \to p} \frac{ x^2 - 4}{x + 3} = \frac{ p^2 - 4}{p + 3} = f(p) . \]

But \( p \) was just a random point in the domain, so the function is continuous on its whole domain, or in other words, it is continuous on \( ( -\infty, -3) \cup (-3, \infty ) \) .

But the process you did in the previous example would work for any rational function. So you can write the following theorem.

**Theorem: ** Every rational function is continuous on its domain.

- Let be an interval in the domain of the function . We say that is
**continuous on the interval \( I \)**if all of the following are true:- \( f(x) \) is continuous at all interior points of \( I \);
- if the left endpoint \( a \) of the interval \( I \) is in the interval, then \( f(x) \) is continuous from the right at \( a \); and
- if the right endpoint \( b \) of the interval \( I \) is in the interval, then \( f(x) \) is continuous from the left at \( b \).

- A function is said to be
**continuous on its domain**if it is continuous at every point in its domain. - The intervals where are function is continuous are called the
**intervals of continuity**. - A function that is continuous on the whole real line is called continuous everywhere.

More about Continuity Over an Interval

60%

of the users don't pass the Continuity Over an Interval quiz! Will you pass the quiz?

Start QuizBe perfectly prepared on time with an individual plan.

Test your knowledge with gamified quizzes.

Create and find flashcards in record time.

Create beautiful notes faster than ever before.

Have all your study materials in one place.

Upload unlimited documents and save them online.

Identify your study strength and weaknesses.

Set individual study goals and earn points reaching them.

Stop procrastinating with our study reminders.

Earn points, unlock badges and level up while studying.

Create flashcards in notes completely automatically.

Create the most beautiful study materials using our templates.

Sign up to highlight and take notes. It’s 100% free.