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Real Analysis: Criteria for Integrability

Suppose the continuous function f:[a,b]->R has the property that:

The integral from c to d f<=0 whenever a<=c<d<=b

Prove that f(x)<=0 for all x in [a,b]. Is this true if we require only integrability of the function?

Solution Preview

Suppose that f(x)>0 for some k such that a<=k<=b

Then by laws of continuity

lim h-->0 f(k+h)>0 and lim h->0 f(k-h)>0

This means that we can always find an ...

Solution Summary

Integrability isused to prove a functional inequality.