Image of countable and uncountable sets 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.
There are non-constant continuous functions with an uncountable number of zeroes.

My first thought was to use the Cantor set $C$, and a search for prior art led to this existing example already on this site:

Non-constant continuous function having uncountably many zeros?

The function is:

$$f:[0,1]\to\mathbb{R}, f(x) = \inf_{c \in C}\{ |x - c|\}$$

It can be shown that $f(x) = 0 \quad\forall x \in C$ and that $f$ is continuous on $[0,1$].
To 1, you dont need suppose continuous.

To 2, as the function is not constant, use intermediate value Theorem and get an interval in the image.