Riemann Integrable

A bounded function is Riemann integrable on iff . In that case, the Riemann integral of on is denoted by .

Reimann Integrable or not

Function	Yes or No?	Proof hint
Unbounded	No	By definition
Constant	Yes
Monotonically increasing/decreasing	Yes	Take a partition such that
Continuous	Yes	Take a partition such that

Note

If the set of points of discontinuity of a bounded function is finite, then is Riemann integrable on .

Note

If the set of points of discontinuity of a bounded function is finite number of limit points, then is integrable on .

A function may have infinitely many discontinuous points, but if the set of all discontinuous points have finite number of limit points, then is integrable on .