True or false: A function $f$ defined on an interval $I$ is continuous if the image of each interval $I_0 \subset I$ is an interval

Question

I think that if $f$ is continuous, the it must map intervals to intervals because intervals are compact in $\mathbb{R}$, and continuous functions map compact sets into compact sets. I suspect the other way around is not true, i.e., there can exists a non-continuous function that maps intervals to intervals. I tried thinking of what one such function might look like, but I'm having trouble thinking of how to map intervals that contain a discontinuity

Answer 1

False. Consider $f$ defined by $f(x) = \sin(1/x)$ when $x \neq 0$, and $f(0)=0$. It's discontinuous at 0, but maps intervals to intervals.

The Conway base 13-function is even more interesting. It is discontinuous everywhere but maps intervals to intervals.

Check out: https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis)