Can a Function Be Integrable but Not Continuous?

Continuous functions are integrable, but continuity is not a necessary condition for integrability. As the following theorem illustrates, functions with jump discontinuities can also be integrable.

Can non continuous functions be integrable?

Is every discontinuous function integrable? No. ... It's not integrable! For any partition of [0,1], every subinterval will have parts of the function at height 0 and at height 1, so there' no way to make the Riemann sums converge.

Can a function be non continuous?

; and second, the function (as a whole) is said to be continuous, if it is continuous at every point. A function is said to be discontinuous (or to have a discontinuity) at some point when it is not continuous there. These points themselves are also addressed as discontinuities. and so for all non-negative arguments.