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Q. 10

Expert-verified
Found in: Page 119

Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

Each function in Exercises 9–12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f. $\underset{x\to {2}^{-}}{\mathrm{lim}}f\left(x\right)=2,\underset{x\to {2}^{+}}{\mathrm{lim}}f\left(x\right)=1,f\left(2\right)=1.$

The type of discontinuity is a jump discontinuity and f(x) is right continuous at $x=2.$

The graph of f is

See the step by step solution

Step 1. Given Information.

The given function is $\underset{x\to {2}^{-}}{\mathrm{lim}}f\left(x\right)=2,\underset{x\to {2}^{+}}{\mathrm{lim}}f\left(x\right)=1,f\left(2\right)=1.$

Step 2. Describing the discontinuity.

From the function, we can depict that $\underset{x\to {2}^{-}}{\mathrm{lim}}f\left(x\right)=2\mathrm{and}\underset{\mathrm{x}\to {2}^{+}}{\mathrm{lim}}\mathrm{f}\left(\mathrm{x}\right)=1$ both exist but are not equal to $f\left(2\right)=1.$

Thus, f(x) has a jump discontinuity at $x=2.$

Step 3. Describing one-sided continuity at x=c.

The $f\left(x\right)$ is right continuous at $x=2$ but not left continuous because $\underset{x\to {2}^{+}}{\mathrm{lim}}f\left(x\right)=f\left(2\right).$

Step 4. Graph of f.

The graph of f is