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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.

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 [[differentiable function]] is always continuous, but a continuous function is not always differentiable. For example, the function <math>f(x) = |x|</math> is continuous everywhere but not differentiable at <math>x = 0</math>.  A more extreme example is the Weierstrass function, which is continuous everywhere but is differentiable only on a measure zero set.

 

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 convergent [[sequence]], [[net]], or [[filter]] to a convergent sequence, net, or filter, respectively.