What does discontinuity mean in math?

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=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=1/x# has no y-value for #x=0#.

In a similar way, the periodic function #y=tanx# has discontinuities at #x=pi/2, (3pi)/2, (5pi)/2...#

Infinite discontinuities occur in rational functions when the denominator equals 0. #y=tan x=(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=((x-3)(x+2))/(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.