The variation of order m of a (deterministic) function f on an interval [a, b] is defined as the limit Prove the following: (a) If f is a bounded function with zero variation of order m on [a, b],...

The variation of order m of a (deterministic) function f on an interval [a, b] is defined as the limit

Prove the following:

(a) If f is a bounded function with zero variation of order m on [a, b], then f has zero variation of order m + 1 on [a, b].

(b) If f is continuous with bounded mth variation on [a, b], then f has zero variation of order k > m on [a, b].

(c) If f has a bounded first derivative on [a, b], then it has bounded first variation and zero variation of order m ≥ 2 on [a, b].

(d) The function
 has bounded first variation and zero variation of order m ≥ 2 on [0, 1].