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iyiswad9k 2022-04-30 Answered
Let I = [ a , b ] with a < b and let u : I R be a function with bounded pointwise variation, i.e.
V a r I u = sup { i = 1 n | u ( x i ) u ( x i 1 ) | } <
where the supremum is taken over all partition P = { a = x 0 < x 1 < . . . < x n 1 < b = x n }. How can I prove that if u satisfies the intermediate value theorem (IVT), then u is continuous?

My try: u can be written as a difference of two increasing functions f 1 , f 2 . I know that a increasing function that satisfies the (ITV) is continuous, hence, if I prove that f 1 , f 2 satisfies the (ITV) the assertion follows. But, is this true? I mean, f 1 , f 2 satisfies (ITV)?
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Answers (1)

Waylon Padilla
Answered 2022-05-01 Author has 19 answers
Begin by proving that every function of bounded variation has finite one-sided limits at every point. (Decomposition into monotone functions does this in one line.) Then observe that the intermediate value property fails unless f ( a + ) = f ( a ) = f ( a ).
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