What makes a function continuous at a point?

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Answer 1 · 2018-05-11T19:04:50

Let $f(x)$ be a function defined in an interval $(a,b)$ and $x_0 in (a,b)$ a point of the interval.

Then the definition of continuity is that the limit of $f(x)$ as $x$ approaches $x_0$ equals the value of $f(x)$ in $x_0$ .

In symbols:

$lim_(x->x_0) f(x) = f(x_0)$

Based on the formal definition of limit, then, for every number $epsilon > 0$ we can find $delta_epsilon > 0$ such that:

$abs(x-x_0) < delta_epsilon => abs(f(x)-f(x_0)) < epsilon$

This means that as $x$ gets closer and closer to $x_0$ also $f(x)$ gets closer and closer to $f(x_0)$ and thus the function is "smooth".