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As the title suggests, I want to find an example where a sequence of continuous functions $\{f_n\}$ converges uniformly on compact sets to a continuous function $f$, and yet the convergence is not uniform over the whole domain. Thank you in advance!

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Take $f_n:(0,1)\to \mathbb R$, with $f(x)=x^n$.

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$x^n$ on $[0,1)$: the compact subsets are contained in $[0,r]$ for some $r<1$.

Other examples are power series with infinite radius of convergence.

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The functions $g_n(x) = \frac{x^n}{n!}e^{-x}$ are another example. They have the property that $\displaystyle \lim_{n\to\infty} g_n =0$, but you can check that $\displaystyle \int_0^\infty g_n dx = 1$ for all $n$.

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Even simpler, try $f_n(x)=\frac xn$ and $f(x)=0$, defined for all real $x$.

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