TITLE: Image of countable and uncountable sets
QUESTION [5 upvotes]: There are two questions:


Image of a countable set of real numbers under any continuous function is countable?


My claim is yes. Let $X$ is countable $\implies X=\{x_1,x_2,\ldots,\}$. Now $f(X)=\{f(x_1),f(x_2),\ldots,\}$ which can be atmost countable. Now my question is "What is the role of continuity here?"


Image of a uncountable set of real numbers under any non-constant continuous function is uncountable?


I feel this is true. But unable to proceed. Please provide me a hint.

REPLY [4 votes]: For 1., you do not need continuity.
For $2$, the function $f: (-1, 0)\cup (0,1)$ defined as $$\begin{cases}-1 & x<0\\
1 & x>0\end{cases}$$
is continuous on $(-1,0)\cup(0,1)$ and not constant.