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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?

UnsinkableSam
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    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
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    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
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    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?) – UnsinkableSam Jun 13 '19 at 11:59
  • In our case case it means closed and bounded. Now $(0,1)$ is bounded, but not closed. – Michael Hoppe Jun 13 '19 at 12:37

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