Continuous function

Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers.

As an example, the function H(t) denoting the height of a growing flower at time t would be considered continuous.

In contrast, the function M(t) denoting the amount of money in a bank account at time t would be considered discontinuous since it "jumps" at each point in time when money is deposited or withdrawn.

[5] Eduard Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854.

There are several different definitions of the (global) continuity of a function, which depend on the nature of its domain.

In other contexts, mainly when one is interested in their behavior near the exceptional points, one says they are discontinuous.

Using mathematical notation, several ways exist to define continuous functions in the three senses mentioned above.

In detail this means three conditions: first, f has to be defined at c (guaranteed by the requirement that c is in the domain of f).

As neighborhoods are defined in any topological space, this definition of a continuous function applies not only for real functions but also when the domain and the codomain are topological spaces and is thus the most general definition.

In proofs and numerical analysis, we often need to know how fast limits are converging, or in other words, control of the remainder.

set) – and gives a rapid proof of one direction of the Lebesgue integrability condition.

Cauchy defined the continuity of a function in the following intuitive terms: an infinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable (see Cours d'analyse, page 34).

In other words, an infinitesimal increment of the independent variable always produces an infinitesimal change of the dependent variable, giving a modern expression to Augustin-Louis Cauchy's definition of continuity.

However, unlike the previous example, G can be extended to a continuous function on all real numbers, by defining the value

The intermediate value theorem is an existence theorem, based on the real number property of completeness, and states: For example, if a child grows from 1 m to 1.5 m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25 m. As a consequence, if f is continuous on

The extreme value theorem states that if a function f is defined on a closed interval

(or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists

Roughly speaking, a function is right-continuous if no jump occurs when the limit point is approached from the right.

This is the same condition as continuous functions, except it is required to hold for x strictly larger than c only.

[14] A function is Hölder continuous with exponent α (a real number) if there is a constant K such that for all

[15] The Lipschitz condition occurs, for example, in the Picard–Lindelöf theorem concerning the solutions of ordinary differential equations.

A topological space is a set X together with a topology on X, which is a set of subsets of X satisfying a few requirements with respect to their unions and intersections that generalize the properties of the open balls in metric spaces while still allowing one to talk about the neighborhoods of a given point.

An extreme example: if a set X is given the discrete topology (in which every subset is open), all functions

In several contexts, the topology of a space is conveniently specified in terms of limit points.

This motivates the consideration of nets instead of sequences in general topological spaces.

This characterization remains true if the word "filter" is replaced by "prefilter.

If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is a homeomorphism.

into all topological spaces X. Dually, a similar idea can be applied to maps

Various other mathematical domains use the concept of continuity in different but related meanings.

A fundamental result known as the Stepanov-Denjoy theorem states that a function is measurable if and only if it is approximately continuous almost everywhere.

The function is continuous on its domain ( ), but is discontinuous at when considered as a partial function defined on the reals. [ 7 ]
The sequence exp(1/ n ) converges to exp(0) = 1
Illustration of the ε - δ -definition: at x = 2 , any value δ ≤ 0.5 satisfies the condition of the definition for ε = 0.5 .
The failure of a function to be continuous at a point is quantified by its oscillation .
The graph of a cubic function has no jumps or holes. The function is continuous.
The graph of a continuous rational function . The function is not defined for The vertical and horizontal lines are asymptotes .
The sinc and the cos functions
Plot of the signum function. It shows that . Thus, the signum function is discontinuous at 0 (see section 2.1.3 ).
Point plot of Thomae's function on the interval (0,1). The topmost point in the middle shows f(1/2) = 1/2.
A sequence of continuous functions whose (pointwise) limit function is discontinuous. The convergence is not uniform.
For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph so that the graph always remains entirely outside the cone.
Continuity at a point: For every neighborhood V of , there is a neighborhood U of x such that