# Continuity

Continuity will usually only be in question when we are dealing with piecewise-defined functions. For these, we say: ”$f$ is continuous at a value $a$ if and only if $f(x)→f(a)$ as $x→a$”

### Types of Discontinuities

Removable discontinuity: We can obtain a continuous function simply by defining $f(a)=c$

Ex: $xsinx $ is a removable discontinuity, we can get it continuous by defining $f(0)=1$

Infinite discontinuity: Approaches $∞$ or $−∞$ as $x→a$

Jump discontinuity: Jumps in points. $lim_{x→a−}f(x)=lim_{x→a+}f(x)$

### Differentiability

$f_{′}(a)=x→alim x−af(x)−f(a) $