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Derivatives and Continuity

- 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
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- 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
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- 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
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- HL ASA and AAS
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- Mechanics Maths
- Acceleration and Time
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- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
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- Converting Units
- Force as a Vector
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- Newton's First Law
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- Projectiles
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- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Probability and Statistics
- Bar Graphs
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- 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
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- 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
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- Chain Rule
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- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
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- Complex Numbers
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- Composition of Functions
- Compound Interest
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- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
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- 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
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- Disproof by Counterexample
- Distance from a Point to a Line
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- Equation of Line in 3D
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- 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
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- Fractions
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- 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
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- Inverse Matrices
- Inverse and Joint Variation
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- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
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- Location of Roots
- Logarithm Base
- Logic
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- Math formula
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- Modulus Functions
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- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
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- Parabola
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- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
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- Proof
- Proof and Mathematical Induction
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- Proportion
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- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
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- Similar Triangles
- Similar and Congruent Shapes
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- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
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- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
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- Special Products
- Standard Form
- Standard Integrals
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- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
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- Surds
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- Tangent of a Circle
- The Quadratic Formula and the Discriminant
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- Transformations of Graphs
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- Triangle Rules
- Triangle trigonometry
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- Trigonometric Functions of General Angles
- Trigonometric Identities
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- 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
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- Bivariate Data
- Box Plots
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- 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
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- Data Analysis
- Data Interpretation
- Discrete Random Variable
- Distributions
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- Errors in Hypothesis Testing
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- 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

Continuing from our Derivatives article, we know that not only are derivatives one of the most important concepts we can learn in Calculus but also that we can write the derivative of a function as a function itself! There is no need to take the derivative of a function at every point where we need to know it; we can find the derivative function!

But what makes a function differentiable? Well, we know that a function is considered differentiable if its derivative exists at every point in its domain. What does this mean, exactly?

This means that a function is differentiable wherever its derivative is defined.

In other words, as long as we can find the derivative at every point on the graph, the function is differentiable.

So, how do we determine if a function is differentiable?

We use Limits and Continuity!

To start, we recall that the **limit **of a function is defined as:

Say we have a function, \( f(x) \), that is defined at all values in an open interval that contains \( a \), (with the possible exception of itself). Let \( L \) be a real number. If **all values** of the function \( f(x) \) approach the real number \( L \) as the values of \( x \) (as long as \( x \neq a \)) approach the number \( a \), then we say that:

- The
**limit**of \( f(x) \) as \( x \) approaches \( a \) is \( L \).- In other words, as \( x \) gets closer and closer to \( a \), \( f(x) \) gets closer and stays close to \( L \).

- This is symbolically expressed as:

\[ \lim_{x \to a} f(x) = L \]

And, the definition of **Continuity **is:

A function, \( f(x) \), is **continuous **at a point, \( p \), if and only if **all of the following are true**:

- \( f(p) \) exists.
- \( \lim_{x \to p} f(x) \) exists.
- The
**limits**from the left and right side of the function at that point are equal.

- The
- \( \lim_{x \to p} f(x) = f(p) \).

If a function fails any of these conditions, then \( f(x) \) is **not continuous** (also called **discontinuous**) at point \( p \).

The expression “if and only if” is a biconditional logic statement. It means that if A is true, then B is also true, and if B is true, then A is also true.

The **Derivative **of a function is defined as:

Say we have a function, \( f(x) \). Its **derivative**, denoted by \( f'(x) \), is the function whose domain contains the values of \( x \) such that the following **limit **exists:

\[ f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \]

Finally, the definition of **differentiability **is:

A function, \( f(x) \), is **differentiable **on an open interval, \( (a, b) \), if the limit,

\[ \lim_{h \to 0} \frac{f(c+h)-f(c)}{h} \]

exists for every number, \( c\), in the open interval, \( (a, b) \).

- If \( f(x) \) is
**differentiable**, meaning \( f'(c) \) exists, then \( f(x) \) is**continuous**at \( c \) in the**open interval**of \( (a, b) \).

As we can see from these definitions, **limits**, **continuity**, and **derivatives **are intertwined. We use limits to:

define continuity and derivatives, and

determine whether functions are continuous and/or differentiable.

Together, these definitions tell us that differentiability is when the slope of the tangent line to the curve equals the limit of the derivative of the function at a point.

- This suggests to us that for a function to be differentiable, it must be continuous, and its derivative must be continuous as well.

We have just stumbled upon a key implication here: **differentiable functions are continuous**.

What does this mean?

Differentiable means that at every point in its domain, the derivative exists for a function.

The only way for the derivative to exist is if the function is continuous on its domain.

Therefore, a differentiable function must also be a continuous function.

This brings us to the theorem – **differentiability**** implies continuity**.

Let \( f(x) \) be a function with \( a \) in its domain. If \( f(x) \) is differentiable at \( a \), then it is also continuous at \( a \).

Proof of the theorem – **differentiability**** implies continuity**.

If \( f(x) \) is differentiable at \( x = a \), then \( f'(a) \) exists and the following limit exists:

\[ f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a} \]

Remember, this is the definition of the derivative!

To prove this theorem, we need to show that \( f(x) \) is continuous at \( x = a \) by showing that \( \lim_{x \to a} f(x) = f(a) \). So,

- Begin with\[ \lim_{x \to a} (f(x) - f(a)) \]
- Multiply and divide by \( (x - a) \) to get\[ \lim_{x \to a} \left( (x - a) \cdot \frac{f(x)-f(a)}{x-a} \right) \]
- Use the product law for limits.\[ \left( \lim_{x \to a} (x - a) \right) \left( \lim_{x \to a} \frac{f(x)-f(a)}{x-a} \right) \]
- Evaluate the first limit and rewrite the second limit as \( f'(a) \) since it is the definition of the derivative.\[ 0 \cdot f'(a) \]
- Therefore,\[ 0 \cdot f'(a) = 0 \]
- This tells us that\[ \lim_{x \to a} (f(x) - f(a)) = 0 \]

- Use the difference law for limits on the equation from step 4b.\[ \lim_{x \to a} f(x) - \lim_{x \to a} f(a) = 0 \]
- Since \( a \) is a constant, simplify the second limit to get\[ \lim_{x \to a} f(x) - f(a) = 0 \]
- Add \( f(a) \) to both sides.\[ \lim_{x \to a} f(x) = f(a) \]

Since \( f(a) \) is defined and \( \lim_{x \to a} f(x) = f(a) \), we can conclude that \( f(x) \) is continuous at \( a \).

So, if differentiability implies continuity, can a function be differentiable but not continuous?

The short answer is no. Just because a function is continuous doesn't mean its derivative is defined everywhere in its domain.

Let's look at the graph of the absolute-value function:

\[ f(x) = |x| \]

We know the absolute-value function is continuous because we can draw the graph without picking up our pencil.

However, we can also see that the slope of the graph is different on the left and right-hand sides. This suggests to us that the instantaneous rate of change (the derivative) is different at the vertex.

So, is the function differentiable here?

- Let's try to take the derivative at \( x = 0 \) and find out. We can do this by taking one-sided limits, using the definition of the derivative to determine if the slope on the left and right sides are equal.

The limit of \( f(x) \) at \( x = 0 \) from the left-hand side of the graph is:

\[ \begin{align}\lim_{h \to 0^{-}} \frac{f(x+h)-f(x)}{h} &= \lim_{h \to 0^{-}} \frac{(-(0+h))-0}{h} \\&= \lim_{h \to 0^{-}} \frac{-h}{h} \\&= \lim_{h \to 0^{-}} (-1) \\&= -1\end{align} \]

The limit of \( f(x) \) at \( x = 0 \) from the right-hand side of the graph is:

\[ \begin{align}\lim_{h \to 0^{+}} \frac{f(x+h)-f(x)}{h} &= \lim_{h \to 0^{+}} \frac{(0+h)-0}{h} \\ &= \lim_{h \to 0^{+}} \frac{h}{h} \\&= \lim_{h \to 0^{+}} (1) \\&= 1\end{align} \]

Comparing these two limits, we see that the slope of the left is \( -1 \) and the slope of the right side is \( 1 \). Because the two limits do not agree, the limit does not exist.

Therefore, the absolute-value function, \( f(x) = |x| \), even though it is continuous, is not differentiable at \( x = 0 \).

This was just one example where continuity does not imply differentiability. A summary of situations where a continuous function is not differentiable include:

If the limit of the slopes of the tangent lines to the curve on the left and right are not the same at any point, the function is not differentiable.

In the case of the absolute-value function, this resulted in a sharp corner of the graph at \(x=0 \). This leads us to conclude that for a function to be differentiable at a point, it must be “smooth” at that point.

A function is not differentiable at any point where it has a tangent line that is vertical.

A function may fail to be differentiable in more complicated ways, such as a function whose oscillations become increasingly frequent as it approaches a value.

So, how can we determine if a function is differentiable?

The quickest way to tell if a function is differentiable is to look at its graph. If it does not have any of the conditions that cause the limit to be undefined, then it is differentiable. These conditions are:

Sharp point

Vertical tangent (where the slope is undefined)

Discontinuity (jump, removable, or infinite)

What are the differences between continuous and differentiable functions?

Continuity is a weaker condition than differentiability.

For a function \( f(x) \) to be continuous at \( x = a \), the only requirement is that \( f(x)-f(a) \) converges to \( 0 \) as \( x \to a \).

For a function \( f(x) \) to be differentiable at \( x = a \), \( f(x)-f(a) \) must converge after being divided by \( x-a \).

In other words, \( \frac{f(x)-f(a)}{x-a} \) must converge as \( x \to a \).

If a function \( f(x) \) is differentiable at \( x = a \), then it is continuous at \( x = a \) as well.

However, if a function is continuous at \( x = a \), it is not necessarily differentiable at \( x = a \).

If \( \frac{f(x)-f(a)}{x-a} \) converges, the numerator converges to zero, which implies continuity.

Spotting where a function is not differentiable.

At which values of \( x \) is \( f(x) \) not differentiable? Why?

**Solution**:

This function is not differentiable in several places. From left to right, these are:

- At \( x = -8 \) there is a Jump Discontinuity.
- At \( x = -6.5 \) there is a vertical tangent.
- At \( x = -4 \) there is a sharp point.
- At \( x = 0 \) there is an infinite discontinuity.
- At \( x = 2 \) there is another sharp point.
- At \( x = 3 \) there is a Removable Discontinuity.

Is the function below continuous and differentiable at \( x = 0 \)?

\[f(x) =\begin{cases}1 & x \lt 0 \\x & x \geq 0\end{cases}\]

**Solution**:

First, let's graph this function.

1. Is the function continuous at \( x = 0 \)?

- To determine if the function is continuous at \( x = 0 \), we can either:
- look at the graph of the function, or
- evaluate the limit as \( x \to 0 \) of both parts of the function.

- Let's start by evaluating the limit of both parts of the function.

\[ f(x) = \lim_{x \to 0^{-}} 1 = 1 \]

\[ f(x) = \lim_{x \to 0^{+}} x = 0 \]

We take One-Sided Limits of the parts of the functions because that is where those parts of the function are valid.

Because we get different results when evaluating the limits, we know the function is **not continuous at** \( \bf{x = 0} \). And, if we look at the graph of the function above, we can confirm that the function is not continuous there.

2. Is the function differentiable at \( x = 0 \)?

Since the function is not continuous at \( x = 0 \), it is also **not differentiable at **\( \bf{x = 0} \).

Is the function below continuous and differentiable at \( x = 4 \)?

\[f(x) =\begin{cases}x^{2} & x \lt 4 \\5x-4 & x \geq 4\end{cases}\]

**Solution**:

First, let's graph this function.

1. Is the function continuous at \( x = 4 \)?

- To determine if the function is continuous at \( x = 4 \), we can either:
- look at the graph of the function, or
- evaluate the limit as \( x \to 4 \) of both parts of the function.

- Let's start by evaluating the limit of both parts of the function.

\[ f(x) = \lim_{x \to 4^{-}} x^{2} = \lim_{x \to 4^{-}} (4)^{2} = 16 \]

\[ f(x) = \lim_{x \to 4^{+}} (5x-4) = \lim_{x \to 4^{+}} (5(4)-4) = 16 \]

We take One-Sided Limits of the parts of the functions because that is where those parts of the function are valid.

Because we get the same result when evaluating the limits, we know the function **is ****continuous at** \( \bf{x = 4} \). And, if we look at the graph of the function above, we can confirm that the function is continuous there.

2. Is the function differentiable at \( x = 4 \)?

To determine if the function is differentiable at \( x = 4 \), we need to use the formula for the definition of a derivative at a point:

\[ f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a} \]

In the same way we took the limit of both parts of the function to test continuity, we will need to take the derivative of both parts of the function to test differentiability.

For the function to be differentiable at \( x = 4 \), the derivative of both parts of the function need to not only exist, but also be equal. In other words:

\[ f'_{-} (a) \mbox{ must be equal to } f'_{+} (a) \]

To take the derivative from the left, we plug in \( x^{2} \) for \( f(x) \), \( 16 \) for \( f(a) \), and \( 4 \) for \( a \) and solve.

\[\begin{align}f'_{-}(a) = \lim_{x \to a^{-}} \frac{f(x)-f(a)}{x-a} & = \lim_{x \to 4^{-}} \frac{x^2-16}{x-4} \\& = \lim_{x \to 4^{-}} \frac{(x+4)(x-4)}{x-4} \\& = \lim_{x \to 4^{-}} (x+4) \\& = 4+4 \\f'_{-}(a) & = 8\end{align}\]

To take the derivative from the right, we plug in \( 5x-4 \) for \( f(x) \), \( 16 \) for \( f(a) \), and \( 4 \) for \( a \) and solve.

\[\begin{align}f'_{+}(a) = \lim_{x \to a^{+}} \frac{f(x)-f(a)}{x-a} & = \lim_{x \to 4^{+}} \frac{5x-4-16}{x-4} \\& = \lim_{x \to 4^{+}} \frac{5x-20}{x-4} \\& = \lim_{x \to 4^{+}} \frac{5(x-4)}{x-4} \\f'_{+}(a) & = 5\end{align}\]

Therefore, while both limits exist, the function is **not differentiable at **\( \bf{ x = 4 } \) because the limit from the left is not equal to the limit from the right. And, if we look carefully at the graph of the function, we can confirm this discontinuity because there is a sharp point at \( x = 4 \).

- The limit of a function is expressed as: \( \lim_{x \to a} f(x) = L \)
- A function is continuous at point \( p \) if and only if all of the following are true:
- \( f(p) \) exists.
- \( \lim_{x \to p} f(x) \) exists, i.e., the limits from the left and right are equal.
- \( \lim_{x \to p} f(x) = p \).

- The definition of the derivative is the limit:
- \( f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \)

- We use Limits to:
define continuity and derivatives, and

determine whether functions are continuous and/or differentiable.

**Differentiability implies continuity, but continuity does not imply differentiability**.- To tell if a function is differentiable, look at its graph. If it does not have any of the conditions that cause the limit to be undefined, then it is differentiable. These conditions are:
sharp points,

vertical tangents,

discontinuities (jump, removable, infinite)

No. If a function is differentiable at a point, it must also be continuous there.

However, a function that is continuous at a point need not be differentiable at that point. In fact, a function may be continuous at a point and fail to be differentiable at the point for one of several reasons.

This brings us to the theorem: differentiability implies continuity.

- Note, however, that the reverse is not true: continuity does not imply differentiability.

Continuity implies differentiability, but differentiability does not imply continuity.

More about Derivatives and Continuity

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