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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 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.