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Continuous

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A property of a function.

Definition. A function f: I\to\mathbb R, where I is a real interval, is continuous in the point a\in I, if for any \varepsilon>0 there exists a number \delta (depending on \varepsilon) such that for all x\in I\cap (a-\delta, a+\delta) -\{a\} we have |f(x)-f(a)| < \varepsilon.

A function is said to be continuous on an interval if it is continuous in each of the interval's points.

An alternative definition using limits is \lim_{x\to a} f(x) = f(a).

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