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In Mathematics , a continuous function is a Function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be '''discontinuous'''. A continuous function with a continuous Inverse Function is called '''bicontinuous'''.

An intuitive though imprecise (and inexact) idea of continuity is given by the common statement that a continuous function is a function whose graph can be drawn without lifting the chalk from the blackboard.

The context of this introduction is one in which the inputs and outputs of functions are Real Number s. More technical definitions are needed for Complex Number s or more general Topological Space s. In Order Theory , especially in Domain Theory , one considers a notion derived from this basic definition, which is known as Scott Continuity .

As an example, consider the function ''h''(''t'') which describes the Height of a growing flower at time ''t''. This function is continuous. In fact, there is a dictum of Classical Physics which states that ''in nature everything is continuous''. By contrast, if ''M''(''t'') denotes the amount of money in a bank account at time ''t'', then the function jumps whenever money is deposited or withdrawn, so the function ''M''(''t'') is discontinuous.


REAL-VALUED CONTINUOUS FUNCTIONS

Suppose we have a function that maps Real Number s to real numbers and whose Domain is some Interval , like the functions ''h'' and ''M'' above. Such a function can be represented by a Graph in the Cartesian Plane ; the function is continuous if, roughly speaking, the graph is a single unbroken Curve with no "holes" or "jumps".

To be more precise, we say that the function ''f'' is continuous at some Point ''c'' when the following two requirements are satisfied:
  • ''f''(''c'') must be defined (i.e. ''c'' must be an element of the Domain of ''f'').

  • The Limit of ''f''(''x'') as ''x'' approaches ''c'' must exist and be equal to ''f''(''c''). (If the point ''c'' in the domain of ''f'' is not a Limit Point of the domain, then this condition is Vacuously True , since ''x'' cannot approach ''c''. Thus, for example, every function whose domain is the set of all integers is continuous, merely for lack of opportunity to be otherwise. However, one does not usually talk about continuous functions in this setting.)


We call the function everywhere continuous, or simply '''continuous''', if it is continuous at every point of its Domain . More generally, we say that a function is continuous on some Subset of its domain if it is continuous at every point of that subset. If we simply say that a function is continuous, we usually mean that it is continuous for all real numbers.

The notation ''C''(Ω) or ''C''0(Ω) is sometimes used to denote the set of all continuous functions with domain Ω. Similarly, ''C''1(Ω) is used to denote the set of differentiable functions whose derivative is continuous, ''C''&2(Ω) for the twice-differentiable functions whose second derivative is continuous, and so on. In the field of computer graphics, these three levels are sometimes called ''g0'' (continuity of position), ''g1'' (continuity of tangency), and ''g2'' (continuity of curvature). The notation C^{(n,\alpha)}(\Omega) occurs in the definition of a more subtle concept, that of Hölder Continuity .


Cauchy definition (epsilon-delta)

Without resorting to limits, one can define continuity of real functions as follows.

Again consider a function ''f'' that maps a set of Real Numbers to another set of real numbers, and suppose ''c'' is an element of the domain of ''f''. The function ''f'' is said to be continuous at the point ''c'' if (and only if) the following holds: For any number \epsilon > 0 however small, there exists some number \delta > 0 such that for all ''x'' in the domain with c - \delta < x < c + \delta, the value of ''f''(''x'') satisfies f(c) - \epsilon < f(x) < f(c) + \epsilon.