Continuous function

In calculus, a continuous function at point x=c is a function whereby 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 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 in Y, the inverse image of Y 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.