Uniform vs Pointwise Convergence

Both definitions say "$f_n(x)$ gets close to $f(x)$ for large $n$". The difference is about how large $n$ has to be. Pointwise: pick a point $x$, then ask how big $n$ must be — the required $n$ can depend on which $x$ you picked. Uniform: ask for one $n$ that works simultaneously for every $x$ in the domain. Geometrically: in pointwise convergence, each vertical slice $\{x = x_0\}$ converges, but the rate may slow down as $x$ varies; in uniform convergence, the entire graph of $f_n$ eventually fits inside the $\eps$-tube around the graph of $f$. The order of quantifiers — $\forall x\, \exists N$ vs $\exists N\, \forall x$ — is the entire content.

The tube is the set of all points within vertical distance $\eps$ of the limit function $f$ — i.e., the strip $\{(x,y) : \abs{y - f(x)} \leq \eps\}$. Uniform convergence is the statement: for every $\eps > 0$, eventually the whole graph of $f_n$ lives inside this tube. The simulation lets you set $\eps$ and watch: for sequences like $\sin(nx)/n$, the curves slip into the tube quickly. For $x^n$ on $[0,1]$, however small you make $\eps$, near $x = 1$ the function $x^n$ stubbornly pokes its head outside the tube no matter how large $n$ gets — because $f(1) = 1$ but $f_n(x)$ is close to 0 for $x$ slightly less than 1. The "uniform verdict" readout reports the empirical sup-norm trend.

Anywhere you need to swap a limit with another operation. Fourier series: convergence of partial sums to a target function is uniform on intervals where the target is smooth, but only pointwise (or even worse) at jump discontinuities — that's where the Gibbs phenomenon lives. Power series: converge uniformly on compact subsets of the disk of convergence, which is why you can integrate and differentiate term-by-term safely. Numerical PDE: error analysis of finite-element/finite-difference schemes asks whether the discrete solution converges uniformly to the true solution — pointwise convergence is too weak for engineering guarantees. Probability: weak vs strong laws of large numbers, and Glivenko–Cantelli (uniform convergence of empirical CDFs to the true CDF) is the foundation of statistical learning theory. Functional analysis: the right notion of "function" depends on which convergence you take seriously — $L^p$, $L^\infty$, $C([a,b])$ are all defined by different convergence modes.

The proof of "continuity is preserved" relies on splitting an estimate into three pieces of size $\eps/3$ — and one of them, $\abs{f(x) - f_N(x)}$, must be small uniformly in $x$ in a neighbourhood of $x_0$. Pointwise convergence only gives smallness at $x_0$, not in a neighbourhood. The classical example is $f_n(x) = x^n$ on $[0,1]$: at every $x_0 \in [0,1)$, you can choose $N$ so $f_N(x_0) < 0.01$. But for $x$ slightly less than 1 (say $x_0 + \delta$), $f_N(x_0 + \delta)$ may still be near 1. So $f_N$ has a sharp transition near 1 that pointwise data at $x_0$ can't see. The limit "function" $f$ inherits that transition as a jump, and continuity fails. Uniform convergence forbids the rate of approach from depending on $x$ — that's exactly what's needed.

Compute $M_n := \supnorm{f_n - f} = \sup_{x \in E}\abs{f_n(x) - f(x)}$ and watch its behaviour as $n$ grows. (a) Analytic: maximize $\abs{f_n - f}$ over $E$ via calculus — set the derivative to zero, find critical points. (b) Numerical: sample, say, $2000$ uniformly-spaced points in $E$, evaluate $\abs{f_n(x_k) - f(x_k)}$ at each, take the max. The simulation does this. Verdict: $M_n \to 0 \Rightarrow$ uniform; $\liminf M_n > 0 \Rightarrow$ not uniform; $M_n$ steadily decaying without reaching 0 ⇒ likely uniform but slowly. The shape of $M_n$ vs $n$ on a log plot is the diagnostic — flat tails are a red flag, while a clean linear decay (in log-log or log-linear) confirms uniform convergence with that rate.

Dini's theorem: if $f_n \to f$ monotonically (i.e., $f_n(x) \leq f_{n+1}(x)$ for all $x$, or the reverse) and pointwise on a compact set $E$, with each $f_n$ and $f$ continuous, then $f_n \to f$ uniformly. The remarkable thing: pointwise convergence usually doesn't promote to uniform, but with these hypotheses (monotone, compact, continuous), it does. Proof uses compactness: open cover by sets where $\abs{f_n - f} < \eps$ admits a finite subcover; monotonicity propagates the bound globally. Counter-example showing each hypothesis is essential: drop compactness and consider $f_n(x) = \min(x, n)$ on $\R$ — monotone increasing to $f(x) = x$, pointwise but not uniform. Drop monotonicity: see $x^n$ on $[0,1]$. The theorem is one of the few places that converts pointwise into uniform — usually you have to assume uniform.

The moving-spike sequence $f_n(x) = n^2 x(1-x^2)^n$ has pointwise limit 0 — at every fixed $x$, $f_n(x) \to 0$. But $\int_0^1 f_n(x)\,dx = \tfrac{n^2}{2(n+1)} \to \infty$. The reason: the peak of $f_n$ moves toward 0 and grows in height like $n^2$, while shrinking in width like $1/\sqrt{n}$, so the area is roughly $n^2 \cdot 1/\sqrt{n} = n^{3/2}$. At each fixed $x > 0$, eventually the spike has moved past $x$, so $f_n(x) \to 0$. But at any moment, there's a tall narrow spike somewhere, contributing area. Integration sees the entire graph at once, not one $x$ at a time — that's why pointwise convergence is too weak. Uniform convergence forbids tall narrow spikes (they'd violate $\supnorm{f_n - 0}$ being small), so it preserves integrals. This is the prototype for why $L^1$ convergence and pointwise convergence are different.