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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. – Dylan Moreland Dec 6 '11 at 4:21

Well, the pointwise limit of the sequence of functions $f_n: [0,1] \rightarrow \mathbb{R}$ given by $f_n(x) = x^n$ is the function which is $0$ on $[0,1)$ and $1$ at $1$. This function is discontinuous only at $x = 1$, so it is not an example of what you're looking for.

Here's a suggestion for a correct answer: try building a sequence by subdividing $[0,1]$ into $n$ pieces and having $f_n$ do something like your sequence of functions at $n$ different points $0,\frac{1}{2},\ldots,1-\frac{1}{2^n}$. You should be able to construct a sequence where the limit function is zero except at points $1-\frac{1}{2^n}$.

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It's easier if you start with a function discontinuous infinitely often and then find continuous functions converging to it. I recommend letting the limiting function be the one that is zero on the irrationals and $1/q$ at the rational $p/q$ (in lowest terms).

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