If is an even continuous function defined on and all its midpoint Riemann sums are zero ( i.e for every ), then is ?
Let be an absolutely summable sequence, and use it to define a function as
It is evident that is even. By the Weierstrass M-Test converges uniformly on to . Hence, is continuous, and using
the fact that we have that
Next, we make two observations:
- If for every then by the definition of the Riemann integral we have that
which yields .
- Using the identity
we obtain the following string of bi-implications for every
In order for all of the mid-point Riemann sums of to be zero, it is thus
necessary and sufficient that
For this reason we choose the sequence as follows
Finally, is continuous , even , non zero and its Riemann mid-points are zero. To see that is non zero we simply calculate ;