Question

a. Let f(x) be a function satisfying |f(x)| <= x^2 for -1 <= x <= 1. Show that f is differentiable at x = 0 and find f'(0).
b. Show that
f(x)=\left\{\begin{array}{ll}{x^{2} \sin \frac{1}{x},} & {x \neq 0} \\ {0,} & {x=0}\end{array}\right.
f(x)={
x
2
sin
x
1
​
,
0,
​

x

=0
x=0
​

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

Answer 1

vertical text and arcane formatting controls are rather off-putting.
Just type it all as you did in the first line.

Answer 2

The definition of differentiable at x=0 means that f'(0) exists.
But f'(0) cannot exist, because f(0) does not exist.

This article may help

math.stackexchange.com/questions/1527146/why-cant-sin1-x-be-differentiated