# Let $f: \Bbb R \to \Bbb R$ be a differentiable function such that $\sup_{x \in \Bbb R}|f'(x)| \lt \infty$. Then

(UGC CSIR-2015, DECEMEMBER, MATHEMATICAL SCIENCES)

1. $f$ maps a bounded sequence to a bounded sequence.
2. $f$ maps a Cauchy sequence to a Cauchy sequence.
3. $f$ maps a convergent sequence to a convergent sequence.
4. $f$ is uniformly continuous.

I choose all of the options as possible answers because the condition $\sup_{x\in \Bbb R}|f'(x)| \lt \infty$ forces $f$ to be uniformly continuous.(Because $f$ becomes Lipschitz and Lipschitz condition implies uniform continuity)

i.e. $\frac {|f(x)-f(y)|}{|x-y|} \le \sup_{x\in \Bbb R}|f'(x)|$ $\forall x,y$.

Hence all other options are bound to be true.

Am I correct?

• correct. The condition even implies that $f$ is Lipschitz. – user295959 Dec 21 '15 at 8:41