a) If is continuous on a closed nondegenerate interval a, b , differentiate on (a,b), and if ' is bounded on (a, b), prove that is of bounded variation on a, b b) Show that (x) 3x is of bounded variation on 0, 1 but ' is unbounded on (0, 1) ...

a Repeat the proof of Remark 551 ...

a) If is continuous on a closed nondegenerate interval [a, b], differentiate on (a,b), and if

Question:

a) If ϕ is continuous on a closed nondegenerate interval [a, b], differentiate on (a,b), and if ϕ' is bounded on (a, b), prove that ϕ is of bounded variation on [a, b].
b) Show that ϕ(x) = 3√x is of bounded variation on [0, 1] but ϕ' is unbounded on (0, 1).