Let f(x) be integrable on [a,b], and let g(x) be nondecreasing and continuously differentiable on [a,b]. Let {p be element of P} be a partition of [a,b], and define

U(f,g,p) = SIGMA (Mi(g(the ith term of x) - g(the (i-1)th term of x))) as i=1 to n
L(f,g,p) = SIGMA (Ni(g(the ith term of x)-g(the (i-1)th term of x))) as i=1 to n

Use mean value theorem to prove that (inf U(f,g,p), for p is element of P) = (sup L(f,g,p), for p is element of P) = ( INTEGRAL f(x)g'(x)dx, as x from a to b)

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Let f(x) be integrable on [a,b], and let g(x) be nondecreasing and continuously differentiable on [a,b]. Let {p be element of P} be a partition of [a,b], and define

Use mean value theorem to prove that

Proof. Given a partition P: of [a,b]. Since g(x) is nondecreasing and continuously differentiable on ...

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