Elementary question on Sobolev spaces and continuity

Question

If a function $\;f \in {W_{loc}}^{1,2} (\mathcal Ω)\;$where $\;\mathcal Ω \;$ a bounded-or not- interval of $ \mathbb R \;$, then is it possible to claim that $\;f\;$ is also continuous there?

If this doesn't hold, what are the sufficient conditions in order to be true the above statement?

Thanks in advance!!!

Answer 1

You have to be a little careful about asserting continuity. If $I$ is any compact subinterval of $\Omega$, then $f \in W^{1,2}(I)$ and in particular $f$ (has a representative that) is absolutely continuous on $I$. It follows that $f$ is equal almost everywhere to a function that is continuous on $\Omega$.