($C^1$ Function)False or true? (justify) a) Let $f: X \to \mathbb R$. If $f'(x) = 0, \forall x \in X$, then $f'$ is constant...

Question

False or true? (justify)

a) Let $f: X \to \mathbb R$. If $f'(x) = 0, \forall x \in X$, then $f(x)$ is constant.

b) If $f$ is differentiable, then $f$ is of class $C^1$. In the case of true prove, in the case of false, provide an example.

a) True, f is derivable when $f'(a) = \lim_{x \to a} \frac{f(x) – f(a)}{x-a}$ is a function and constant if and only if its derivative is equal to zero at all points in the range. As from the statement we have that $f'(x) =0$ for all x belonging to X, so X is constant.

b) False. $f(x) = x^2 sin (\frac{1}{x^2})$, if $x \ne 0$ and $0$ if $x = 0$.

I know a is true and b is false but I'm not sure if I've shown this properly.

Thank's for any help.

Answer 1

a) is False.

Let $$X=(-\infty,0)\cup (0,+\infty)$$

and $ f $ defined at $ X $, by

$$(\forall x<0)\;\; f(x)=-1$$ and $$(\forall x>0)\;\;f(x)=1$$

$ f $ is differentiable at $ X $ and

$$(\forall x\in X)\;\; f'(x)=0$$ But $ f $ is not constant.

b) is also false. Your counterexample is good.