Concept

Differentiability

The property of having a tangent at a point, meaning the chord slopes settle on a single value as the two points come together. It is strictly stronger than continuity, and it is what licenses replacing a curve by a straight line near a point.

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

Secants closing on the tangent to x². Secant lines through x = 1 and a second point 1.2, 0.8, 0.5, 0.28, 0.12 away, with the slope of each. They approach 2, the derivative there.

The slope of a single point

A slope needs two points. A derivative is the slope at one. The construction that bridges the gap is a sequence of secants, and the whole difficulty of calculus is in what "the limit of that sequence" is allowed to mean.

analysis · The derivative

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

A curved map of the plane, and the flat one that fits it at a point. The map (x² − y², 2xy) carrying a small square patch of grid. Beside it, the image of the same patch under the linear map given by the matrix of partial derivatives, drawn dashed on top of the curved image.

The flat map that fits closest

A derivative is usually met as a number, which works because a line through a point is described by one. In more than one dimension the object that plays the same role is a linear map, and the number was always a one-by-one instance of it.

analysis · The derivative

Riemann's function on one period, with its one kind of smooth point. R(x) = Σ sin(n²x)/n² on [0, 2π]; R(π) = 0 with derivative −1/2; R(π/2) = 1.2336, a cusp.

The slope a Gauss sum leaves behind

Riemann is said to have offered sin x + sin(4x)/4 + sin(9x)/9 + … as a continuous function with no derivative anywhere. It has one after all, at π and at every π times an odd number over an odd number, and the slope there is always exactly −1/2. Computed with a million terms, the function's behaviour at every fraction is read off a single number — a quadratic Gauss sum — and the slope appears exactly where that number is zero.

analysis · Uniform convergence

Named alongside it

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

Derivative Continuity Difference quotient Fourier series Limit Approximation Basis Convergence Counterexample Dense set Determinant Fractal dimension

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