Types of Discontinuities

$f(x) = \frac{1}{x}$ Discontinuity at $x=0$
$f(x) = \frac{x^2-9}{x-3}$ Removable Discontinuity at $x=3$
$f(x) = \left\{ \begin{array}{ll} x^2, & x < 0 \\ 1, & x=0 \\ x+1, & x>0 \end{array} \right.$ Discontinuity at $x=0$
$f(x) = \left\{ \begin{array}{ll} 1, & x<0 \\ 3, & x\geq 0 \end{array} \right.$ Jump Discontinuity at $x=0$
$f(x) = \sin \left( \frac{1}{x} \right)$ Discontinuity at $x=0$