Verify the function it is uniformly convergent or not?

Question

Let $f:\mathbb R\rightarrow \mathbb R$ be a continous function . Which of the following statesment are true?

1. If $f$ is bounded, then f is uniformly continous .

2. If $ f $ is differentiable and $f'$ is bounded, then $ f$ is uniformly continous.

3. If $\lim_{|x|\rightarrow \infty}f(x)$ = $0$, then f is uniformly continous.

For (1) is false because we take $f(x) = Sin(x^3)$ is continous and bounded, but it is not uniformly continous.

For (2) is true

For (3) I think it is false , but i am not sure

Thank you for sparing your valuable time in checking my solutions

Answer 1

You're right about the first two, but wrong about the third one. Consider the following argument:

Suppose $f \to 0$ as $|x| \to \infty$, and let $\epsilon>0$ be given. We may select an $r_0>0$ so that whenever $|x|>r_0$, $|f(x)| < \epsilon/2$.

We know that $f$ is uniformly continuous on $[-r_0,r_0]$ because this is a compact interval. We know that $f$ is uniformly continuous on $(-\infty,r_0)$ and $(r_0,\infty)$. Finally, we note that $f$ is continuous at $\pm r_0$.

Using these three facts, we may find a $\delta>0$ so that $|x-y|<\delta$ implies $|f(x) - f(y)|< \epsilon$.