Uniform Convergence

A sequence of functions $f_n$ converges pointwise to a limit function $f$ if, at each individual point $x$, the numbers $f_n(x)$ eventually get close to $f(x)$ — but how quickly "eventually" arrives is allowed to be different at every single point, with no shared speed limit across the whole domain. Uniform convergence asks for something stronger: that there's a single $N$ (a number of steps) after which $f_n$ is close to $f$ at every point simultaneously, not just eventually close at each point separately with a possibly different $N$ for each.

Formally, $f_n \to f$ uniformly on a set means for every tolerance $\epsilon > 0$, there exists one $N$ such that for all $n > N$ and all $x$ in the domain at once, $|f_n(x) - f(x)| < \epsilon$. Pointwise convergence only requires that $N$ to exist for each $x$ separately, and it's allowed to grow arbitrarily large as $x$ changes. The classic example distinguishing them is $f_n(x) = x^n$ on $[0,1]$: it converges pointwise to a function that's $0$ everywhere except $1$ at $x=1$, but not uniformly, because getting close to $0$ near $x=1$ takes an $N$ that grows without bound as $x$ approaches $1$ — there's no single $N$ that works everywhere.

The reason uniform convergence gets its own name rather than being a footnote is that several properties that "should" pass to the limit only actually do so under uniform, not merely pointwise, convergence: a uniform limit of continuous functions is continuous, and a uniform limit lets you swap limits with integrals. Pointwise convergence alone can silently break both — the $x^n$ example converges to a discontinuous limit despite every $f_n$ being continuous, precisely because the convergence isn't uniform.