Concept

Oscillation

The oscillation of a function at a point is how far its values still spread in arbitrarily small neighbourhoods of that point. In the limit it is the gap between the largest and smallest nearby values, and it is zero exactly where the function is continuous.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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

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

The slope of x² sin(1/x): discontinuous at nought, and never jumping. The derivative of x^2 sin(1/x) on [−0.12, 0.12], with the value at nought marked and the level 0.5 crossed 77 times.

A slope can swing but never jump

A function can have a slope at every point without that slope changing continuously: x² sin(1/x) has slope nought at the origin and a slope that swings between −1 and 1 however close to the origin it is taken. What a slope cannot do is jump. Darboux proved in 1875 that a derivative takes every value between any two of its values, so a step is never a derivative — and the only way a slope can be discontinuous is by oscillating.

analysis · The derivative

Oscillators scattered below the threshold and gathered above it. N = 240, Lorentzian width 0.5: K = 0.5 gives r = 0.068; K = 2.5 gives r = 0.775.

A crowd of clocks that falls into step

Give a thousand oscillators a thousand different natural rhythms and let each be pulled towards the average phase. Below a definite pull nothing happens; above it a cluster forms and grows, and Kuramoto's 1975 calculation says exactly where: at twice the spread of the frequencies, with the order growing as the square root of one minus the threshold over the pull. A simulation lands on that curve to the third decimal.

dynamics · Sensitive dependence

Named alongside it

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

Continuity Counterexample Baire category Bifurcation Cantor set Central limit theorem Derivative Intermediate value theorem Limit Mean value theorem Nowhere dense Phase transition

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