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continuous function
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Diagram illustrating a topological mapping from set X to set Y, with a point p in subset U of X mapped to f(p) in subset f(U) of Y, showing the concept of continuity in topology. Figure 2: Piecewise-linear approximations to a continuous function (see text). intermediate value theorem Figure 4: Continuous and discontinuous functions.
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continuous function

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Diagram illustrating a topological mapping from set X to set Y, with a point p in subset U of X mapped to f(p) in subset f(U) of Y, showing the concept of continuity in topology.
The topological concept of a continuous function

A function f from a topological space X to a topological space Y is continuous at p ∊ X if, for any neighbourhood V of f(p), there exists a neighbourhood U of p such that f(U) ⊆ V.

In mathematics, a continuous function is one that varies without abrupt breaks or jumps. Intuitively, if the input values (x) of a function are close together, then the output values (y) will also be close together.

More formally, a function f(x) is continuous at a point x0 if for any desired closeness of y-values (\u03b5), there's a corresponding closeness of x-values (\u03b4) such that if x is within \u03b4 of x0, then f(x) is within \u03b

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4 Britannica Sources
  1. continuity
  2. mathematics - Making the calculus rigorous
  3. analysis - Functions
  4. analysis - Continuity of functions

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