# What does discontinuity mean in math?

Aug 14, 2014

A function has a discontinuity if it isn't well-defined for a particular value (or values); there are 3 types of discontinuity: infinite, point, and jump.

Many common functions have one or several discontinuities. For instance, the function $y = \frac{1}{x}$ is not well-defined for $x = 0$, so we say that it has a discontinuity for that value of $x$. See graph below.

Notice that there the curve does not cross at $x = 0$. In other words, the function $y = \frac{1}{x}$ has no y-value for $x = 0$.

In a similar way, the periodic function $y = \tan x$ has discontinuities at $x = \frac{\pi}{2} , \frac{3 \pi}{2} , \frac{5 \pi}{2.} . .$

Infinite discontinuities occur in rational functions when the denominator equals 0. $y = \tan x = \frac{\sin x}{\cos x}$, so the discontinuities occur where $\cos x = 0$.

Point discontinuities occur where when you find a common factor between the numerator and denominator. For example,
$y = \frac{\left(x - 3\right) \left(x + 2\right)}{x - 3}$
has a point discontinuity at $x = 3$.

Point discontinuities also occur when you create a piecewise function to remove a point. For example:
f(x)={x, x!=2; 3, x=0}
has a point discontinuity at $x = 0$.

Jump discontinuities occur with piecewise or special functions. Examples are floor, ceiling, and fractional part.