Pointwise vs Uniform Convergence — 2D Trace View

Function Sequence

Index n 1

Epsilon band ε 0.150

Convergence Test

—

sup|fₙ−f| (this n)

—

Smallest N for this ε

—

Loading…

How it works

A sequence of functions f_n: D → ℝ converges pointwise to f if, at every fixed x, the number sequence f_n(x) → f(x). It converges uniformly if the worst-case gap over the whole domain vanishes:

Pointwise: ∀x ∀ε>0 ∃N(x, ε): n≥N ⟹ |fₙ(x)−f(x)| < ε
Uniform:   ∀ε>0 ∃N(ε): n≥N ⟹ |fₙ(x)−f(x)| < ε  for ALL x at once
Equivalently: uniform  ⟺  sup_x |fₙ(x) − f(x)| → 0

This 2D view splits the same statement into two linked plots instead of one 3D surface. The top plot is a plain Cartesian graph: the current curve f_n(x) against x, the dashed limit function f(x), and a shaded ε-band hugging f(x) — points on the curve outside the band are drawn red, points inside are green. The bottom plot is the quantity that actually decides uniform convergence: the sup-norm S(n) = sup_x|f_n(x) − f(x)| plotted against n itself, with a dashed horizontal line at the chosen ε. Uniform convergence means that trace eventually drops below the line and stays there for every larger n; non-uniform convergence means it never does, no matter how far n runs.

xⁿ on [0,1] — each f_n is continuous, but the pointwise limit is 0 on [0,1) and jumps to 1 at x=1: discontinuous. Since a uniform limit of continuous functions must be continuous, this sequence cannot converge uniformly — the sup-norm trace sits flat at 1 forever.
sin(nx)/n on [0,2π] — |f_n(x)| ≤ 1/n everywhere, so the sup-norm trace itself decays like 1/n toward 0: uniform convergence to f = 0.
n·x·(1−x)ⁿ on [0,1] — a "travelling bump" that slides toward x = 0 as n grows. At every fixed x the value still → 0 (pointwise), and the limit function 0 is perfectly continuous — yet the sup-norm trace flattens out near 1/e ≈ 0.368 and never decays: not uniform, even though continuity alone didn't rule it out.

This is the central subtlety of real analysis courses: pointwise limits can lose properties (continuity, integrability, differentiability) that every term in the sequence had; uniform convergence — visible directly as the sup-norm trace hitting zero — is exactly the extra condition that preserves them.

fₙ(x) inside ε-band fₙ(x) outside ε-band ε-band around limit f(x) sup-norm trace S(n)