Continuous function

In mathematics, a continuous function is a function where small changes in the input cause only small changes in the output. This means the function's value does not jump or change suddenly. There are no abrupt changes, called discontinuities, in a continuous function.

Until the 19th century, mathematicians used simple ideas about continuity and mostly studied functions that behaved this way. Later, they created a precise way to define continuity using the epsilon–delta definition of a limit.

Continuity is very important in calculus and mathematical analysis, where inputs and outputs are real or complex numbers. The idea of continuity has also been expanded to many different types of functions, which helps build the field of topology.

For example, the height of a growing flower over time can be modeled by a continuous function. But the amount of money in a bank account changes suddenly when money is added or taken out, so it is a discontinuous function.

History

The idea of a continuous function was first described by Bernard Bolzano in 1817. Later, Augustin-Louis Cauchy explained it by saying that tiny changes in the input of a function should only cause tiny changes in the output. In the 1830s, Bolzano and Karl Weierstrass worked on a more exact way to describe continuity, though Bolzano's work was not shared until much later. Different mathematicians, including Édouard Goursat and Camille Jordan, suggested slightly different rules for what makes a function continuous. Eduard Heine gave the first published explanation of uniform continuity in 1872, based on ideas from Peter Gustav Lejeune Dirichlet.

Real functions

A real function that works with real numbers can be shown as a line on a graph. This function is continuous if the line does not have any sudden jumps or breaks.

We usually check for continuity by looking at how small changes in the input affect the output. If small changes in input lead to small changes in output, the function is continuous. This means there are no sudden jumps in the values of the function.

Related notions

When we talk about a continuous function, we mean a function where small changes in the input lead to small changes in the output. There are different ways this idea is used in math.

One important idea is called a continuous extension. This is when we have a function that works on part of a bigger set, and we can expand it to work on the whole set while still being continuous.

In other areas of math, like order theory or category theory, the idea of continuity is used in different but related ways. For example, in order theory, a function is continuous if it preserves certain limits. In category theory, a functor is continuous if it commutes with small limits.

There are also special types of continuity, like approximate continuity, used in measure theory. This is when a function is continuous almost everywhere, meaning it is continuous except for a very small set of points.

Main article: Tietze extension theorem Main article: Hahn–Banach theorem Main article: Hausdorff space Main article: dense subset Main article: Blumberg theorem Main article: order theory Main article: partially ordered sets Main article: directed subset Main article: supremum Main article: Scott topology Main article: category theory Main article: functor Main article: categories Main article: limits Main article: class Main article: diagram Main article: objects Main article: quantales Main article: domains Main article: measure theory Main article: Lebesgue measurable set Main article: approximately continuous Main article: approximate limit Main article: Stepanov-Denjoy theorem