Determine if the graph of the function $f\left(x\right)$is continuous and/or differentiable at $x=1$.

b. Show that

f(x) = { x² sin(1/x), x ≠ 0

0, x = 0

is differentiable at x = 0 and find f′(0).

Theory and Examples

In Exercises 51 and 52, give reasons for your answers.

Let f(x) = |x³ − 9x|.

a. Does f'(0) exist?

Determine if the graph of the function $f\left(x\right)$is continuous and/or differentiable at $x=2$.

Determine if the graph of the function $f\left(x\right)$is continuous and/or differentiable at $x=1$.

Determine if the function$f\left(x\right)$ is continuous and/or differentiable at $x=3$.

Determine if the function$f\left(x\right)$ is continuous and/or differentiable at $x=2$.

Determine where the function $f\left(x\right)$ is not differentiable.

$f\left(x\right)=\frac{3}{x+2}$

Determine if the function$f\left(x\right)$ is continuous and/or differentiable at $x=3$.