  ### Home > CALC3RD > Chapter Ch5 > Lesson 5.2.4 > Problem5-97

5-97.

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

1. $f ( x ) = \left\{ \begin{array} { l l } { 2 x ^ { 3 } } & { \text { for } x < 0 } \\ { 3 x ^ { 2 } } & { \text { for } x \geq 0 } \end{array} \right.$

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

To test if the function is continuous at the boundary point, use the 3 Conditions of Continuity:

$1. \lim \limits_{x\rightarrow a}f(x)\text{ exist. (This means that the limit from the left equals the limit from the right.)}$

2. $f\left(a\right)$ exists.

$3. \ f(a)=\lim \limits_{x\rightarrow a}f(x)$

To test if the function is differential at the boundary point, use the same 3 conditions on the derivative.

$1. \lim \limits_{x\rightarrow a}f'(x)\text{ exist. (This means that the limit from the left equals the limit from the right.)}$

2. $f ^\prime\left(a\right)$ exists. (Note, differentiability implies continuity.)

$3. \ f'(a)=\lim \limits_{x\rightarrow a}f'(x)$

Notice that part (b) has two boundary points, so you will have to run these tests twice.