If the derivative of a function is zero, is the function a constant function?

Question

If the derivative of a function is zero, is the function then a constant function ?

I think it is not true, because if f in the sub interval be constant function then derivative of $f$ is zero, is it true?

Answer 1

If $f$ is a real function which is continuous in the closed inteval $[a,b]$ suppose that $\forall x\in (a,b):f'(x)=0$ than $f$ is constant on $[a,b]$

Proof:

Let $y\in[a,b]$

Then $f$ satisfies the conditions of the Mean Value Theorem on $[a,y]$

Hence:

$\exists \xi \in (a,y):f'(\xi)=\frac{f(y)-f(a)}{y-a}$

But:

$f'(\xi)=0$

which means:

$f(y)-f(a)=0$

and hence:

$f(y)=f(a)$

as $y$ is any $y\in[a,b]$, the result follows

$\square$

Answer 2

The Cantor function is an example of a function that is continuous (even uniformly continuous, but not absolutely continuous) with zero derivative almost everywhere, but increases from 0 to 1.