Concept

Uniform convergence

Convergence of a sequence of functions in which the largest gap to the limit, taken over the whole domain at once, goes to zero. It is strictly stronger than every point settling separately, and it is what a limit needs in order to inherit continuity from its members.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

6 cosines, and a curve with no tangent anywhere. Partial sums of a sum of cosines whose amplitudes shrink geometrically and whose frequencies grow faster. Each term adds finer detail; the curve converges and its slopes do not.

A curve with a corner at every point

Continuity means a curve can be drawn without lifting the pen. Differentiability means it has a tangent. The first was assumed to nearly imply the second until 1872, when Weierstrass exhibited a curve that is continuous everywhere and has a tangent nowhere — and it is a sum of cosines.

analysis · the derivative

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

Named alongside it

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

Continuity Counterexample Convergence Difference quotient Differentiability Fourier series Fractal dimension Function space Gibbs' phenomenon Limit function Pointwise convergence Self similarity

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