0

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function. Suppose that there is a constant $B$ such that $\left| f^ı\left( x\right) \right| \leq B$ for all $x$ in $\mathbb{R}$. Prove that $f$ is uniformly continuous.

Can you give a hint?

1 Answers1

0

HINT: $\vert f'(x) \vert \leq B$ for all $x \in \mathbb{R}$ implies that $$\left\vert \frac{f(x) - f(y)}{x - y} \right\vert \leq B$$ for all $x,y \in \mathbb{R}$.

Ethan Alwaise
  • 9,286
  • Can we use mean value theorem? –  Apr 03 '17 at 20:14
  • 1
    The mean value theorem tells you that the hint is true. – Ethan Alwaise Apr 03 '17 at 20:15
  • By the mean value theorem, we find $\left| f(x)-f(y)\right| \leq B \left| x-y\right|$ for all $x,y\in\mathbb{R}$. So, how can I show that this is uniformly continuous? –  Apr 03 '17 at 20:24
  • 1
    The answer is staring you in the face. Choose $\epsilon > 0$. Then $$B\vert x - y \vert < \epsilon$$ if $\vert x - y \vert < $... – Ethan Alwaise Apr 03 '17 at 20:30