Difference between revisions of "Continuous function"
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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. | + | 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. | 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. | ||
Revision as of 04:12, December 1, 2007
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.