Example of continuous functions $f\colon S \to T$ such that $f(S)=T$.

Question

I would like to find an example of a continuous function from $S=(0,1)$ to $T=(0,1)\cup (1,2)$ such that $f(S)=(T)$. At the moment the only thing I can think might work would be to check whether $f(S)$ is a compact subset - is this correct? Does anybody know how I might do this? Does such a function even exist? Thank you!

Answer 1

There is no such map. Other wise there'd be some $a,b\in S$ with $f(a)=\frac12$ and $f(b)=\frac32$. Then by the IVT, there exists $c$ between $a$ and $b$ (hence $c\in S$) with $f(c)=1\notin T$.

The obstacle has nothing to do with compactness (and neither $S$ nor $T$ is compact), but with connectedness: $S$ is connected and $T$ is not.