Example for continuous function on the metric space R

Question

I am looking for an example for a continuous function $f: R \rightarrow R$, with $(R,d)$ the real line with the standard metric, such that there is an open set $V$ but $f(V)$ is not open.

I tried $f(x)=2^x$. I could found a closed set with its image not closed. But I could not for an open set.

So I need some help or hints about that. Thank you.

Answer 1

Take $f(x)=\frac1{1+x^2}$. Then $f(\mathbb{R})=(0,1]$.

Answer 2

A simple example is $f(x) = c$ where $c$ is constant. Then every open set $V \subset \mathbb{R}$ has $f(V)= \{c\}$.

Edit: In general, a continuous function $f$ is an open mapping (that is, whenever $V$ is open, $f(V)$ is open) if and only if it is strictly monotonic.