# What makes a function continuous at a point?

May 11, 2018

Let $f \left(x\right)$ be a function defined in an interval $\left(a , b\right)$ and ${x}_{0} \in \left(a , b\right)$ a point of the interval.

Then the definition of continuity is that the limit of $f \left(x\right)$ as $x$ approaches ${x}_{0}$ equals the value of $f \left(x\right)$ in ${x}_{0}$.

In symbols:

${\lim}_{x \to {x}_{0}} f \left(x\right) = f \left({x}_{0}\right)$

Based on the formal definition of limit, then, for every number $\epsilon > 0$ we can find ${\delta}_{\epsilon} > 0$ such that:

$\left\mid x - {x}_{0} \right\mid < {\delta}_{\epsilon} \implies \left\mid f \left(x\right) - f \left({x}_{0}\right) \right\mid < \epsilon$

This means that as $x$ gets closer and closer to ${x}_{0}$ also $f \left(x\right)$ gets closer and closer to $f \left({x}_{0}\right)$ and thus the function is "smooth".