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 For example, the function <math>f: X -(x) = |x|</math> Y mapping elements in a [[topological space]] X to a topological space Y is continuous if for every [[open set]] U in Y, everywhere but not differentiable at <math>x = 0</math>. A more extreme example is the inverse image of U under f Weierstrass function, which is an open subset of Xcontinuous everywhere but is differentiable only on a measure zero set.
A continuous function maps a convergent [[sequence]], [[net]], or [[filter]] to a convergent sequence, net, or filter, respectively.