Theorems on Variation

Carothers Chapter 13 [#!kn:Carothers2000!#] gives theorems on the variation in one dimension. See especially p203-204.

Theorem 1 (Expected Error)   If h:[a,b]->R is monotonic, then

$$V(h,P) = \vert h(b)- h(a)\vert$$ (2.4)

for any partition P of [a,b]. It follows that the total variation equals $\vert h(b)-h(a)\vert$.