Continuous function

Continuity is a concept central to calculus, advanced calculus and topology. In calculus, continuity is defined based on limits. In advanced calculus, continuity is defined using neighborhoods or sequences. In topology, a function is continuous if the inverse image of every open set in the function's range is also an open set in the function's domain. In all three fields of mathematics, the unifying characteristic of continuity is that points near each other in a set or domain are mapped by the continuous function to points that are near each other in the corresponding set or range.

In calculus, a function f(x) is said to be continuous at point c if f(c) equals the limit of f(x) as x approaches c from both the positive and negative directions.

Another way of understanding this is by recognizing that a discontinuous function over a specific interval is one that has a gap in the interval, or one having different limits at a particular point depending on whether it is approached from the positive or negative directions.

A simple example of a continuous function would be Y = 2X + 5.

An example of a discontinuous function is Y = 1/X, which has no value for X = 0; also the limits of the function as X approaches zero from each side are different.

A differentiable function is always continuous, but a continuous function is not always differentiable.

A function f: X -> Y mapping elements in a topological space X to a topological space Y is continuous if for every open set U in Y, the inverse image of U under f is an open subset of X.

A continuous function maps a convergent sequence, net, or filter to a convergent sequence, net, or filter, respectively.

A continuous function maps a compact space to a compact space.