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?
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 ...
Integrability isused to prove a functional inequality.