What is "continuous function"

n. 1 (context analysis English) a function whose value changes only slightly when its input changes slightly 2 (context analysis topology English) a function from one topological space to another, such that the inverse image of any open set is open

In mathematics, a continuous function is, roughly speaking, a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.

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. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps at each point in time when money is deposited or withdrawn, so the function M(t) is discontinuous.

In mathematics, specifically set theory, a continuous function is a sequence of ordinals such that the values assumed at limit stages are the limits ( limit suprema and limit infima) of all values at previous stages. More formally, let γ be an ordinal, and s :  = ⟨s∣α < γ⟩ be a γ-sequence of ordinals. Then s is continuous if at every limit ordinal β < γ,

s = limsup{s∣α < β} = inf{sup{s∣δ ≤ α < β}∣δ < β} 

and

s = liminf{s∣α < β} = sup{inf{s∣δ ≤ α < β}∣δ < β} .

Alternatively, s is continuous if s: γ → range(s) is a continuous function when the sets are each equipped with the order topology. These continuous functions are often used in cofinalities and cardinal numbers.