Theorem. Let the real function be Riemann-integrable on . If is an
even function, then ,
odd function, then
Of course, both cases concern the zero map which is both
odd and even.
Proof. Since the definite integral is additive with respect to the interval of integration, one has
Making in the first addend the substitution and swapping the limits of integration one gets
which settles the equations of the theorem.