Discontinuity

In this final step of the integrals section we will take a look at how to calculate integrals of discontinuous functions.

All discontinuity points are divided into discontinuities of the first and second kind. The function (f(x)) has a discontinuity of the first kind at (x=a) if:

• There exist left-hand limit (displaystyle lim_{x to a^{-}}f(x)) and right-hand limit (displaystyle lim_{x to a^{+}}f(x))
• These one-sided limits are finite.

Further there may be the following options:

1. The right-hand limit and the left-hand limit are equal to each other: (displaystyle lim_{x to a^{-}}f(x)=displaystyle lim_{x to a^{+}}f(x)). Such a point is called a removable discontinuity.
2. The right-hand limit and the left-hand limit are unequal: (displaystyle lim_{x to a^{-}}f(x)neq displaystyle lim_{x to a^{+}}f(x)). In this case the function (f(x)) has a jump discontinuity.
3. The function (f(x)) is said to have an infinite discontinuity at (x=a) if at least one of the one-sided limits either does not exist or is infinite.

Intuitively, a removable discontinuity is a discontinuity for which there is a hole in the graph, a jump discontinuity is a non-infinite discontinuity for which the sections of the function do not meet up, and an infinite discontinuity is a discontinuity located at a vertical asymptote.

When we have a piecewise function, in order to calculate the integral over one interval we first break up the interval of integration into pieces based on the function definition and then integrate each piece.