Decide if the following statements are true or false and justify yor answer.
(a) Every bounded function $f:\mathbb Q\cap [0,1]\to\mathbb R$ is continuous.
(b) Every continuous function $g:\mathbb Q\cap [0,1]\to\mathbb R$ is uniformly continuous.
(c) Let $h:\mathbb Q\cap [0,1]\to\mathbb R$. The following statements are equivalent:
i) There exists a continuous function $\overline{h}:[0,1]\to\mathbb R$, $\overline{h}|_{\mathbb Q\cap [0,1]}=h$.
ii) $h$ is uniformly continuous.
My solution:
(a) False, let $$f(x)=\begin{cases} 0 & \text{si}& x\leq 1/2\\ 1 & \text{si}& x> 1/2\end{cases}$$. (b) Is true, since $\mathbb Q\cap [0,1]$ is compact and $g$ is continuous.
(c) $i)\Rightarrow ii)$ Since $\overline{h}$ is continuous, then $h$ is continuous, and for (b) $h$ is uniformly continuous.
$i)\Leftarrow ii)$ Since $h$ is unif. cont. it can be extended to an uniformly continuous function.