Let $\;f\in L^2_{loc}(\mathbb R;\mathbb R^m)\;$. I know that $\; L^2(\mathbb R^n)\;$ is the completion of the continuous functions with respect to the $\;L^2$-norm.
I'm wondering if this is enough in order to claim that $\;f\;$ is equal almost everywhere to a continuous function.
From my point of view, since continuous functions are dense in $\;L^2\;$, there must be a sequence of continuous functions converging to a square integrable one.But I'm not very sure if this implies the above.
I 've been stuck to this so any help would be valuable. I apologize in advance if my question is quite silly.
Thanks
"I know that $L^2(\mathbb R^n)$ is the completion of the continuous functions with respect to the $L^2$-norm." Not all continuous functions! You need continuous functions subject to some kind of growth condition. A favorite here is the space of continuous functions with compact support.
"I'm wondering if this is enough in order to claim that $\;f\;$ is equal almost everywhere to a continuous function." No, that is far far away from being true. Consider $f= \chi_{[0,1]}$ for example.