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From wikipedia,

continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.

I can understand what is a continuous function. A function which just don't have any holes or jumps or vertical asymptotes. But how can we prove that a small change of input in continous function causes only small change in the outputs.

What do we define as small? Why cant a function do a big jump at point X and come back gradually around X+4 etc..?

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    $\begingroup$ It can, the Wikipedia article it trying to give some intuition. The point is that by staying sufficiently close to the point (sufficiently close being dependent on the particular function) the value of $f$ will remain close to the value at thpoint in question. $\endgroup$
    – copper.hat
    Oct 15 '17 at 18:47
  • $\begingroup$ The $\epsilon$ - $\delta$ definition of a continous function basically shows it. $\endgroup$
    – Peter
    Oct 15 '17 at 18:48
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Continuous functions are defined that way. You cannot prove it, itś simply what we define a continuous function as. We don't define a continuous function like you say you "know" it, "without any jumps or holes", that's not the formal definition.

Formally, for every $\epsilon > 0$ there exists a $\delta > 0$ such that $|x-y| < \delta$ implies $|f(x) - f(y)| < \epsilon$.

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