Riemann Integral Proof

**jlindbo** · February 15th 2008, 07:24 AM

I am having a bit of trouble with this proof and any insight would be appreciated.

Suppose f is a nonnegative Riemann integrable function on [a,b] satisfying f(r) = 0 for all r in $\textbf{Q} \cap [a,b]$

Prove that $\int_a^b f = 0$

**Plato** · February 15th 2008, 08:17 AM

I will try to help even though the definitions you are using may be different but equivalent.
Recall that between any two numbers there is a rational number.
Thus, given any division of [a,b] then on any subinterval the lower bound is zero.
Thus given $\varepsilon >0$ there is a division of [a,b] $D$ such that $\left| {\overline {\int_D f } - \underline {\int_D f } } \right| = \left| {\overline {\int_D f } } \right| < \varepsilon$
That implies that the integral is zero.

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