A continuous function on an open interval bounded?

I have being taught that a continuous function on a bounded interval is bounded then I am thinking of a counterexample when the interval is open. The example I came up with was $$f(x)=log(x)$$ for $$x \in (0,1)$$, but I am kind of thinking how could I prove its continuity on $$(0,1)$$ from definition?

• Another example is $f(x)=\frac1x$. You have been taught something wrong, or you do not remember it correctly. A continuous function on a closed bounded interval is bounded. – Hagen von Eitzen Jun 12 '19 at 20:05
• To address the second concern, the usual definition of $\log$ is as the inverse function of $\exp$ and the inverse of a monotonic function on an interval is continuous. – Thorgott Jun 12 '19 at 20:12
• Here's an example of demonstration that $log(x)$ is continuous (on $(0, +\infty)$) using the definition. – lurker Jun 12 '19 at 22:05
• You've better have been taught that the image of a compact set unter a continuous function is compact as well. – Michael Hoppe Jun 13 '19 at 10:38
• @MichaelHoppe Ah our course hasn't properly defined what compactness really means, so I am guessing that is no use for now :( (what is compactness anyways?) – JustWandering Jun 13 '19 at 11:59

If $$f$$ is continous in a interval [a,b], and $$I\subset[a,b]$$ is an interval, then $$f|_I$$ is continous.