Continuous function

related topics
{math, number, function}
{day, year, event}
{math, energy, light}
{style, bgcolor, rowspan}
{woman, child, man}

Power rule, Product rule, Quotient rule, Chain rule

Lists of integrals
Improper integrals
Integration by:
parts, disks, cylindrical
, substitution,
trigonometric substitution,
partial fractions, changing order

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".

Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article.

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if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous. (However, if one assumes a discrete set as the domain of function M, for instance the set of points of time at 4:00 PM on business days, then M becomes continuous function, as every function whose domain is a discrete subset of reals is.)


Full article ▸

related documents
Primitive recursive function
Dual space
Euler's formula
Fundamental theorem of algebra
Basis (linear algebra)
BCH code
Hyperreal number
Ackermann function
Fundamental group
Computable number
Multivariate normal distribution
Bessel function
Dynamic programming
Halting problem
Fermat number
Prime number theorem
Lp space
Probability theory
Group action
Abelian group
Subset sum problem
Monte Carlo method
Frame problem
Truth table
Uniform space
Taylor series
Support vector machine