Any function f(x) which is discontinuous at finite number of points in an interval [a, b] can be made continuous in sub-intervals by breaking the intervals into these sub-intervals.
If f(x) is discontinuous at points x1, x2, x3.....xn in (a, b),
then we can define sub-intervals (a, x1), (x1, x2).....(xn – 1, xn),(xn, b)
such that f(x) is continuous in each of these sub-intervals.
Such functions are called piece-wise continuous functions.
For integration of piecewise continuous function,
we integrate f(x) in these sub-intervals and finally add all the values.
Integration of modulus functions are done in this way.
with proper sign of f(x) in each (a, α), (α, β) and (β, b) sub-intervals.