Nonzero derivative implies function is strictly increasing or decreasing on some interval Let $f$ be a differentiable function on open interval $(a,b)$. Suppose $f'(x)$ is not identically zero. Show that there exists an subinterval $(c,d)$ such that $f(x)$ is strictly increasing or strictly decreasing on $(c,d)$.
How to prove this?
I think this statement is wrong...

I also think the statement is not true. There are examples of a function that has derivative positive but it is not monotone in any neighborhood of that point. Consider $\phi(0)=0$ and for $x\neq 0$ consider
$$\phi(x)=x^2 \sin\left(\dfrac{1}{x}\right)+\dfrac{x}{2}$$
This function is continuous, $\phi^\prime(0)>0$ but not monotone near zero
