Let : [0, 1] R be continuously differentiable (i.e., they are differentiable and their derivatives...

  

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Let ₁: [0, 1] → R be continuously differentiable (i.e., they are differentiable and their derivatives are continuous) on (0,1), with a also increasing there. Show that there is c> 0 (o,) with such that for any >0, we have U(P.f.a) - L(P.f.a)< whenever P = 0=< < <= 1 satisfies Az, <ce for all i=1,2,...,n. Let ₁: [0, 1] → R be continuously differentiable (i.e., they are differentiable and their derivatives are continuous) on (0,1), with a also increasing there. Show that there is c> 0 (o,) with such that for any >0, we have U(P.f.a) - L(P.f.a)< whenever P = 0=< < <= 1 satisfies Az, <ce for all i=1,2,...,n.