= Uniform limit theorem
{wiki}
A <uniform limit> of <continuous functions> is a <continuous function>. No <compactness> assumption on the <domain of a function> is required.<Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl/6.2 Interchange of limits>[Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl, §6.2.1, Theorem 6.2.2]{ref}
For a proof, fix an input $x$ and tolerance $\varepsilon>0$. <Uniform convergence> supplies one $f_N$ with $\|f_N-f\|_\infty<\varepsilon/3$. Its <continuity> at $x$ supplies $\delta>0$ with $|f_N(y)-f_N(x)|<\varepsilon/3$ for allowed inputs satisfying $|y-x|<\delta$. The <triangle inequality> gives
$$
|f(y)-f(x)|\leq|f(y)-f_N(y)|+|f_N(y)-f_N(x)|+|f_N(x)-f(x)|<\varepsilon.
$$
One approximating <function> controls both endpoints of this estimate. Under <pointwise convergence>, selecting an index for $x$ supplies no common control at nearby $y$. This explains why the <polynomial functions> $x^n$ can have a discontinuous <pointwise limit> on $[0,1]$.
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