Prove that a function whose derivative is bounded is uniformly continuous.

Question

Suppose that $f$ is a real-valued function on $\Bbb R$ whose derivative exists at each point and is bounded. Prove that $f$ is uniformly continuous.

Answer 1

Since $f'$ is bounded then there's $M>0$ s.t. $$|f'(x)|\leq M\quad \forall x\in\mathbb{R}$$ hence by mean value theorem we find $$|f(x)-f(y)|\leq M|x-y|\quad \forall x,y\in\mathbb{R}$$ so $f$ is a lipschitzian function on $\mathbb{R}$ and therefore it's s uniformly continuous on $\mathbb{R}$.

Answer 2

Hint: $f(a)-f(b)=f'(\xi)(a-b)$.