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Continuous

From AoPSWiki

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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