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I am trying to find a convergent sequence of continuous real-valued functions on $[0,1]$ whose limit function fails to be continuous at an infinite number of points. I have thought about $f_n(x)=x^n$ which is convergent to $1$, but how do I show the limit fails to be continuous at an infinite number of points?

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    Also, I TeX-ified your post and added what I think is a better title. Feel free to edit.2011-12-06

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