What makes a function continuous at a point?

What makes a function continuous at a point?
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

embraci4i
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:
$\underset{x\to {x}_{0}}{lim}f\left(x\right)=f\left({x}_{0}\right)$
Based on the formal definition of limit, then, for every number $\epsilon >0$ we can find ${\delta }_{\epsilon }>0$ such that:
$|x-{x}_{0}|<{\delta }_{\epsilon }⇒|f\left(x\right)-f\left({x}_{0}\right)|<\epsilon$
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".