Continuity
Named by 21 essays across 5 fields — each of them below, with the objects they name alongside it.
A square wave built entirely out of round ones
Add enough sine waves together and flat tops and vertical cliffs appear from nothing. Almost — there is a 9% overshoot that never goes away, and it is not a bug.
The curve that is its own slope
There is exactly one shape of exponential curve whose steepness at every point equals its height at that point. The number that produces it is 2.71828…, and it was not chosen for elegance.
A sphere is a plane plus one point
Remove a single point from a sphere and what is left can be flattened out to cover an infinite plane exactly. The construction is one straight line, repeated.
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.
Something always stays put
Stir a cup of coffee however violently and let it settle. Some molecule is exactly where it started. Crumple a map and drop it on the region it depicts, and one point lies over the place it names.
Area is the undoing of slope
Two operations invented for unrelated reasons — measuring a region and measuring a rate — turn out to be inverse. The picture is two panels sharing one axis, and the claim is that the lower curve's steepness is the upper curve's height.
Nothing on a sphere can be combed flat
Point an arrow along the surface at every place on a sphere, continuously, and somewhere an arrow has to vanish. On a doughnut it can be done. The difference between the two is a number that was already known from counting corners.
A loop that cannot miss the middle
Feed a circle into a polynomial and a closed loop comes out. A small circle gives a loop that does not enclose the origin; a large one gives a loop that goes round it as many times as the degree. Something has to happen in between, and that something is a root.
Which side of the line is inside
A closed curve with no self-crossings divides the plane into an inside and an outside. Nobody doubts it, almost nobody can prove it, and on a curve wound tightly enough nobody can see which side a given point is on either.
The staircase that is not the diagonal
A staircase can be made to follow a quarter circle as closely as anyone likes. Its length is 2 at every stage and the arc's length is 1.5708, and no amount of refinement closes the gap — which is a fact about length rather than about staircases.
One line that halves them both
Two shapes lying anywhere on a page, of any sizes and any shapes at all. There is always a single straight line that cuts both of them into two equal halves at once — and finding it needs no cleverness, only the observation that a quantity which reverses sign has to pass through zero.
A loop that cannot be pulled tight
A hole is a strange thing to point at, because it is precisely where the surface is not. What can be pointed at is a loop of string lying on the surface — and the hole announces itself by refusing to let that loop be pulled in to a point.
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.
The walk that becomes a curve
Shrink the steps of a random walk and it disappears. Shrink them while stretching the time in the right proportion — space by the square root of whatever time is divided by — and something is left behind, which is a curve nobody could draw.
The slope of the mirror image
Undoing a function is reflecting its graph in the diagonal, and a reflection turns a slope into its reciprocal. That single observation supplies the derivative of every inverse — the logarithm, the roots, the inverse trigonometric functions — without differentiating any of them.
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.
A line with as many points as a square
Interleave the decimal places of two numbers and one number comes out; take every other place back and the two return. The square has no more points than the segment, and dimension turns out to be invisible to counting.
A curve that has area
The Jordan curve theorem assumes three things and nothing else — continuous, closed, no self-crossing. Everything else the eye supplies is false of some curve that satisfies all three, including the assumption that a curve is thin.
Every loop is a circle in disguise
Separating the plane is the weak half of what the eye believes about a closed curve. The strong half is that the inside is a disc — that the whole plane can be bent until the curve is a round circle — and for a polygon that is a construction rather than an argument.
Which functions can be added up
Riemann's integral works when the upper and lower sums close on each other. The exact condition for that, found once measure existed to state it in, is that the points where the function jumps have measure zero — which some nowhere dense sets fail.
A staircase with no steps
A function that rises from nought to one, is continuous everywhere, and has derivative zero at almost every point. All of its climbing happens on a set of no length at all, which is possible because that set has uncountably many points.
Named alongside it
The objects these essays reach for when they reach for this one.
LimitCounterexampleDerivativeWinding numberClosed curveNonconstructiveBrouwerCantor setConvergenceFixed pointGibbs' phenomenonHomeomorphism