# Darboux Integrability epsilon-delta proof

• Mar 26th 2013, 05:08 AM
jll90
Darboux Integrability epsilon-delta proof
Hello all, I'm having trouble proving this theorem.

Suppose f is Darboux integrable, then for all $\epsilon > 0$ there exists a $\delta > 0$ such that mesh (P) < $\delta$ implies U_p(f) - L_p(f) < $\epsilon$.

Proof:

Let $\epsilon > 0$, and suppose f is Darboux Integrable.

If f is a constant we are done, if not pick $mesh (P) < \frac{\epsilon}{\sum\limits_{k=1}^n M_k - m_k}$, where k is the nth sub-interval of the partition P. M_k is the supremum of the function in that respective sub-interval, m_k is the infimum.

It follows that,

$\vert U(f) - L(f) \vert < \vert U_p (f) - L_p (f) \vert < \epsilon$.

However, my instructor said that, I cannot pick that delta because the suprema and infima of the sub-intervals depend on the partition. I left out some steps, because the problem with the proof is the choice of delta.

Any thoughts on this? Thank you