If the derivative of a function is $\alpha$-Hoelder continous, is the original function $\alpha$-Hoelder continuous?

Question

Let $I \subseteq \mathbb R$ be some interval and let $\alpha \in (0,1]$.

A function $f : I \rightarrow \mathbb R$ satisfies the Hoelder condition if there exists a constant $C > 0$ such that

$$\forall x,y \in I : |f(x) - f(y)| \leq C |x-y|^\alpha$$

Suppose that $f$ is differentiable and that its derivative $f'$ satisfies the Hoelder condition with exponent $\alpha$. Does $f$ satisfy the Hoelder condition with exponent $\alpha$?