The following statement was provided to me by a senior to be used while solving problems of uniform continuity.
Statement : A function $f(x)$ is uniform continuous on only that interval for which $f'(x)$ is bounded.
Although I have studied analysis from Stephen Abott and Tom Apostol , this result is not given there.
Can you please give hints on how to prove it?
Thanks!!
Theorem: If $f'$ is bounded then $f$ is uniformly continuous.
Proof Suppose $f'$ is bounded then there's $M>0$ s.t. $$|f'(x)|\leq M\quad \forall x\in\mathbb{R}$$ Using mean value theorem we have, $|f(x)-f(y)|\leq M|x-y|\quad \forall x,y\in\mathbb{R}$ Note that $f$ is a lipschitzian function on $\mathbb{R}$ $\Rightarrow$ $f$ uniformly continuous on $\mathbb{R}$.
Converse need not be true, Take $f:\Bbb{R}^+\to \Bbb{R^+}$ such that $f(x)=\sqrt{x}$. $f$ is uniformly continuous but $f'$ is not bounded in $\Bbb{R}^+.$