Compact,connected set

Question

How could be proved that a function that sends compacts into compacts and connected into connected implies f continuous ? I’d appreciate even just the idea behind that (if the proof is too long) Thanks

@ChrisCulter I don't think this duplicates that question. This question is not restricted to $\mathbb R^n$. — Umberto P., Mar 02 '18 at 19:11
@UmbertoP. I'm assuming that the OP is working with some unstated assumptions, since they seem to be looking for a proof of a true statement. — Chris Culter, Mar 02 '18 at 19:24
@ChrisCulter possibly. On the other hand, there are lots of requests on this site for proofs of false statements. — Umberto P., Mar 02 '18 at 19:25
I don’t know if it’s true or false,i’d like to know it (and why) so that i could work on the proof — jacopoburelli, Mar 02 '18 at 19:36

score 0 · Accepted Answer · answered Mar 02 '18 at 19:10

Let $X$ and $Y$ be topological spaces and $f : X \to Y$.

If $X$ is a totally disconnected space its only connected subsets are singletons. In this case $f$ sends connected sets to connected sets.

If $Y$ is a two-point set then every subset of $Y$ is compact. In this case $f$ sends compact sets to compact sets.

So, you should look for a discontinuous function from (say) the irrational real numbers with the relative topology to the set $\{0,1\}$ to obtain a counterexample.

Compact,connected set

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