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 $f(x)\to f(a)$ as $x\to a$”

Types of Discontinuities

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

Ex: $\frac{\sin x}{x}$ is a removable discontinuity, we can get it continuous by defining $f(0)=1$

Infinite discontinuity: Approaches $\infty$ or $-\infty$ as $x\to a$

Jump discontinuity: Jumps in points. ${\lim }_{x\to a-}f(x)\ne {\lim }_{x\to a+}f(x)$

Differentiability

$f^{\prime }(a)=\underset{x\to a}{\lim }\frac{f(x)-f(a)}{x-a}$