Which of the following is not uniformly continuous?

Question

Which of the following is not uniformly continuous?

1.$f_1(x)=|x|$

2.$f_2(x)=\frac{1}{1+x^2}$

3.$f_3(x)=\sin x^2$

4.$f_4(x)=\ln(1+x^2)$

5.$f_5(x)=e^{-x}$

My solution:$f_1(x)=|x|$ is lipschitz so uniformly continuous.

$\lim_{ x\to\pm \infty}\frac{1}{1+x^2}=0$.Also $f_2(x)$ is continuous,so it is uniformly continuous

$f_3(x),f_4(x),f_5(x)$ are not uniformly continuous.Am i right?

Answer 1

A function with bounded derivative is uniformly continuous (see for instance Prove that a function whose derivative is bounded is uniformly continuous.). Hence 4) is uniformly continuous.