# Difference between revisions of "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. | 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. | ||

## Revision as of 04:47, 29 August 2008

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 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.