is there a continuous and open map on topological vector space X into X such that is not onto and is not constant map?

Question

Could anyone give an example for a non constant continuous and open map ‎$f‎ : X\rightarrow X$ such that $f(0)=0$ and that is not onto (X is topological vector space) and $‎\frac{(f(X)+f(X))}{2}‎\nsubseteq‎ ‎f(X)‎$? I suppose there must be some examples, but I wasn't able to find one yet. Thank you for answers.

Answer 1

Take $X = \mathbb{R}$ and $f(x)=\arctan(x)$, this is a continuous bijection from $\mathbb{R}$ to $(-\frac{\pi}{2}, \frac{\pi}{2})$.