### Home > APCALC > Chapter 7 > Lesson 7.2.2 > Problem7-68

7-68.

No calculator! Determine where the following functions are not continuous and/or non-differentiable.

1. $f(x) =\left\{ \begin{array} { l l } { \frac { 1 } { 2 } x ^ { 2 } + \frac { 3 } { 2 } } & { \text { for } x \leq - 1 } \\ { | 3 - x | } & { \text { for } x > - 1 } \end{array} \right.$

Check continuity: Does $\lim\limits_{x\rightarrow -1^{-}}f(x)=\lim\limits_{x\rightarrow -1^{+}}f(x)=f(-1)?$

Check differentiability: Does$\lim\limits_{x\rightarrow -1^{-}}f'(x)=\lim\limits_{x\rightarrow -1^{+}}f'(x)=f'(-1)?$

Also, recall that differentiability implies continuity.

2. $g(m) =\left\{ \begin{array} { l l l } { \frac { 2 } { m - 3 } } & { \text { for } } & { m < 2 } \\ { - \sqrt { 6 - m } } & { \text { for } } & { 2 \leq m < 5 } \\ { \frac { 1 } { 2 } m - \frac { 7 } { 2 } } & { \text { for } } & { m \geq 5 } \end{array} \right.$

Refer to the hints in part (a).