Functions, with Limited Variation.—The condition that ƒ(x), in the mean-value theorem, either never increases or never diminishes as x increases from a to b, places a restriction upon the applications of the theorem. We can, however, show that a function ƒ(x) which is finite and continuous between a and b, except for a finite number of ordinary discontinuities, and which only changes from increasing to diminishing or vice versa, a finite number of times, as x increases from a to b, may be expressed as the difference of two functions ƒ1(x), ƒ2(x), neither of which ever diminishes as x passes from a to b, and that these functions are finite and continuous, except that one or both of them are discontinuous at the points where the given function is discontinuous. Let α, β be two consecutive points at which ƒ(x) is discontinuous, consider any point x1, such that α ≦ x1 ≦ β, and suppose that at the points M1, M2 ... Mr between α and x1, ƒ(x) is a maximum, and at m1, m2 ... mr, it is a minimum; we will suppose, for example, that the ascending order of values is α, M1, m1, M2, m2 ... Mr, mr, x1; it will make no essential difference in the argument if m1 comes before M1, or if Mr immediately precedes x1, Mr−1 being then the last minimum.
Let
ψ(x1) = [ƒ(M1) − ƒ(α + 0)] + [ƒ(M2) − ƒ(m1)] + ... + [ƒ(Mr) − ƒ(mr−1)] + [ƒ(x1) − ƒ(mr)];
now let (x1) increase until it reaches the value (Mr+1) at which ƒ(x) is again a maximum, then let
ψ(x1) = [ƒ(M1) − ƒ(α + 0)] + [ƒ(M2) − ƒ(m1)] + ... + [ƒ(Mr) − ƒ(mr−1)] + [ƒ(Mr+1) − ƒ(mr)];
and suppose as x increases beyond the value Mr+1, ψ(x1) remains constant until the next minimum mr+1 is reached, when it again becomes variable; we see that ψ(x1) is essentially positive and never diminishes as x increases.
Let
χ(x1) = [ƒ(M1) − f(m1)] + [ƒ(M2) − ƒ(m1)] + ... + [ƒ(Mr) − ƒ(mr)],
then let x1 increase until it is beyond the next maximum Mr+1, and then let
χ(x1) = [ƒ(M1) − ƒ(m1)] + [ƒ(M2) − ƒ(m1)] + ... + [ƒ(Mr) − ƒ(mr)] + [ƒ(Mr+1) − ƒ(x1)]