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If a function series $f_n$, where each $f_n(x)$ is continuous, converges point wise to $f$ and $f$ is not continuous, does this imply that $f_n$ does not convergence uniformly to $f$.

I'm pretty sure this is true, but I didn't found a reliable source for it. Would you please help me to find such a source or provide a proof for it.

Additional, does the sentence above also hold if $f_n(x)$ is not continuous?

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up vote 4 down vote accepted

Yes. It's a theorem that if a sequence of continuous functions converges uniformly, then the limiting function is continuous. As the contrapositive, if the limiting function is not continuous, the convergence cannot be uniform.

If the functions $f_n$ are not continuous, then they can certainly converge uniformly to a non-continuous function. (For example, all the $f_n$'s could be the same non-continuous function.)

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