please use advance calculus to prove this problem(problem 10) Suppose that the f

Question

please use advance calculus to prove this problem(problem 10)

Suppose that the functions f:[a, b] rightarrow R and g: [a,b] rightarrow R are integrable. Show that there is a sequence {Pn} partitions of [a,b] that is an Archimedean sequence of partitions for f an [a, b] and also an Archimedean sequence of partitions for g on [a, b]. (Hint: the Refinement Lemma) Prove that the function f: [2,4] rightarrow R is integrable. For a partition P = {x0, ..., xn} of the interval [a, b], show that Suppose that the function f: [a, b] rightarrow R is Lipschitz; that is, that there is a constant c 0 such that for all points u, v in [a,b]. For a partition P of p[a.b], prove that

Explanation / Answer

integrable means y' should be existing at all points i.e x = 3 in this problem

so, f'(x<3) = 1 and f'(x>3) = 0

so, both exists, hence the function is integrable

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