# Thread: Differentiability Proof of two functions

1. ## Differentiability Proof of two functions

Let f,g := $R \rightarrow R$ be given by

f(x) :=
$x sin(\frac{1}{x}), x \neq 0$
0, x = 0

and

g(x) :=
$x^2 sin(\frac{1}{x}), x \neq 0$
0, x = 0

Prove that f is not differentiable at x = 0 and that g is differentiable at x = 0.

2. Originally Posted by thaopanda
Let f,g := $R \rightarrow R$ be given by

f(x) :=
$x sin(\frac{1}{x}), x \neq 0$
0, x = 0

and

g(x) :=
$x^2 sin(\frac{1}{x}), x \neq 0$
0, x = 0

Prove that f is not differentiable at x = 0 and that g is differentiable at x = 0.
A function F(x), is differentiable at x= 0 if and only if the limit $\lim_{h\to 0}frac{F(h)- F(0)}{h}$ exists. Do that for both f(x) and g(x) and see what you get!