CPM Homework Help : CALC Problem 5-97

Sketch each of the following piecewise functions. Then, determine if the functions are continuous and differentiable over all reals.

$f ( x ) = \left\{ \begin{array} { l l } { 2 x ^ { 3 } } & { \text { for } x < 0 } \\ { 3 x ^ { 2 } } & { \text { for } x \geq 0 } \end{array} \right.$
$f ( x ) = \left\{ \begin{array} { c c } { ( x + 1 ) ^ { 2 } + 1 } & { \text { for } } \\ { | x | } & { \text { for } - 2 \leq x < 2 } \\ { \operatorname { sin } ( x - 2 ) + 2 } & { \text { for } } & { x \geq 2 } \end{array} \right.$

Steps:

To test if the function is continuous at the boundary point, use the three conditions of continuity:

$f(a)$ exists.
To test if the function is differential at the boundary point, use the same three conditions on the derivative.
$f^\prime(a)$ exists. (Note, differentiability implies continuity.)
$\ f'(a)=\lim\limits_{x\rightarrow a}f'(x)$
Notice that part (b) has two boundary points, so you will have to run these tests twice.