Continuous function

In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. If small changes in the input can produce a broken jump in the changes of the output (or the value of the output is not defined), the function is said to be discontinuous (or to have a discontinuity). The context in this entry is real-valued functions on the real domain or on topological or metric spaces other than the complex numbers; for complex-valued functions see complex analysis. The notable difference in approach is that in the present context, the points in the domain that would be regarded as singularities (points of discontinuity) in the complex domain are usually assumed to be absent, or they are explicitly excluded, so as to leave a function that is continuous on a disconnected real domain.

Related Topics:
Mathematics - Function - Change - Input - Output - Complex analysis - Domain

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As an example, consider the function h(t) which describes the height of a growing flower at time t. This function is continuous (unless the flower is cut). As another example, if T(x) denotes the air temperature at height x, then this function is also 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if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous.

Related Topics:
Height - Classical physics

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There are also some more special usages of continuity in some mathematical disciplines. Probably the most common one, found in topology, is described in the article on continuity (topology). In order theory, especially in domain theory, one considers a notion derived from this basic definition, which is known as Scott continuity.

Related Topics:
Topology - Continuity (topology) - Order theory - Domain theory - Scott continuity

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