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Continuous Function
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The most common notion of continuity in topology defines continuous functions as those functions for which the preimages of open sets are open. Similar to the open set formulation is the closed set formulation, which says that preimages of closed sets are closed.
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Most functions you normally encounter are either continuous, or else continuous everywhere except at a finite collection of points. For any such function, an antiderivative always exists except possibly at the points of discontinuity.
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Continuous Function