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Could someone give me a hint of how I might find a sequence of uniformly continuous functions that converges only pointwise to a (not uniformly) continuous function? Thanks.

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Yes, but also, the limit function must not be uniformly continuous. – Dan Sep 17 '11 at 19:25
up vote 3 down vote accepted

Let $f_n(x):=\min(n^2,x^2)$. Let $n\in\mathbb N$. $f_n$ is continuous and for $x$ such that $x^2\geq n^2$ we have $f_n(x)=n^2$ hence $f_n$ has a limit when $x\to+\infty$ and $x\to-\infty$, so $f_n$ is uniformy continuous on $\mathbb R$. If we fix $x\in\mathbb R$, we can find $n_0$ such that $f_n(x)=x^2$ if $n\geq n_0$ (for example, take $n_0=\lfloor x\rfloor+1$). We conclude that the sequence $\{f_n\}$ converges pointwise to $x\mapsto x^2$, which is continuous but not uniformly continuous on $\mathbb R$.

We can see the convergence on this graph: Some terms of the sequence.

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First decide what the limit is going to be -- this can be almost anything, for example $f(x)=x^2$, which is continuous but not uniformly continuous on $\mathbb R$.

Then construct a series of uniformly continuous functions that converge pointwise towards your goal. The thing to remember here is that when the convergence is only pointwise, it is allowed to take (much) longer time to converge at some points than at others.

(Hint: a piecewise continuously differentiable function that is constant everywhere but in a finite interval is automatically uniformly continuous).

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