Continuous function

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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.