Concept

Pointwise convergence

A sequence of functions converges pointwise when, at each point separately, the values converge to the limit's value there. It is weaker than uniform convergence, which asks one rate to work everywhere at once, and the gap between them is where continuity, integrals and limits stop being preserved.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

xⁿ at 5 values of n, and the limit. Several members of the sequence xⁿ drawn on one pair of axes with the function they settle on, and the largest gap between each member and that limit reported.

A limit that forgets to be continuous

Every one of the functions x, x², x³, … is as smooth as anything could be, and every column of the picture settles down. What they settle on has a jump in it — and the quantity that sees the difference is the largest gap anywhere, which is a number about the whole graph rather than about any point of it.

analysis · Uniform convergence $Thomae's function as a limit of tents, at n = 3 and 8. Continuous functions built from narrow triangles over the fractions, drawn at two stages, with the limit shown as a dot at height 1/q over every fraction p/q: a function continuous at the irrationals and discontinuous at the rationals.$

A limit can jump at every fraction

A sequence of continuous functions can settle, point by point, on a function that is discontinuous at every rational number. It cannot settle on one that is discontinuous everywhere — the indicator of the rationals needs two limits in a row, and Riemann's integral cannot follow the second. The line between the two is Baire's theorem, and it measures smallness by gaps rather than by length.

analysis · Uniform convergence

xⁿ is uniform off a strip, with N = 14, 29, 59 for 3 strips. The functions x to the n on the unit interval with a band of half-width 0.05 around zero. For each of 3 strips next to 1, the member at which every later one stays in the band away from the strip.

Uniform, except on a small set

The functions xⁿ settle on their limit at every point and never uniformly — the trouble is all in a strip next to 1. Throw the strip away and the convergence is uniform on what is left, however thin the strip. Egorov proved that this always happens on an interval, Lusin proved the matching fact about a single function, and a bump sliding off along the whole line shows why both need a set of finite length to start from.

analysis · Uniform convergence

Polynomials climbing to the square root, each above the last. The iterates of p ↦ p + (x − p²)/2 at steps 1, 2, 3, 4, 6, 9 on [0, 1] beneath the curve √x, with largest gaps 1.000, 0.500, 0.259, 0.176, 0.108, 0.068.

Settling in order settles everywhere at once

A sequence of functions can settle at every point and never settle uniformly — the largest gap stays put while each point escapes it. Dini found the circumstances in which that cannot happen: continuous functions, a continuous limit, every member above the next, and a closed interval. Under those four conditions convergence at each point is convergence everywhere at once, and Weierstrass's iteration for the square root becomes a sequence of polynomials converging uniformly to |x|.

analysis · Uniform convergence

Named alongside it

The objects these essays reach for when they reach for this one.

Continuity Counterexample Uniform convergence Rational number Supremum Baire category Cantor set Compactness Convergence Convergence rate Function space Geometric series

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