Chapter 1.5, Problem 36ES

A function $f$ is said to have a removable discontinuity at $x=c$ if ${\lim }_{x\to c}f\left(x\right)$ exists but $f$ is not continuous at $x=c,$ either because $f$ is not defined at $c$ or because the definition for $f\left(c\right)$ differs from the value of the limit. This terminology will be needed in these exercises.

Find the values of $x$ (if any) at which $f$ is not continuous, and determine whether each such value is a removable discontinuity.

(a) $f\left(x\right)=\frac{x^2-4}{x^3-8}$

(b) $f\left(x\right)=\left\{\begin{array}{cc}2x-3,& x\le 2\\ x^2,& x>2\end{array}\right.$

(c) $f\left(x\right)=\left\{\begin{array}{cc}3x^2+5,& x\ne 1\\ 6,& x=1\end{array}\right.$