If is an even continuous function defined on and all its midpoint Riemann sums are zero ( i.e for every ), then is ?

**Solution**

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(i) and

(ii)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 ;