Proof
Let f satisfy the hypotheses. Let r be any smooth function that is 0 at a and b and positive on (a, b); for example, . Let . Then h is of class on, so
The integrand is nonnegative, so it must be 0 except perhaps on a subset of of measure 0. However, by continuity if there are points where the integrand is non-zero, there is also some interval around that point where the integrand is non-zero, which has non-zero measure, so it must be identically 0 over the entire interval. Since r is positive on (a, b), f is 0 there and hence on all of .
Read more about this topic: Fundamental Lemma Of Calculus Of Variations
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