Sum of increasing and decreasing functions

[chakib]

i want to know if any real function can be expressed as:
f(x)=g(x)+h(x) such as g(x) is an increasing function and h(x) is a decreasing function?
thanks

 

[BvU]

Hello chakib,

Here at PF we try to help folks to help themselves by (mostly) not providing direct answers, but providing help in the form of comments, hints, nudges, etc.

In this case: suppose you succeed, what does that mean for the derivatives of g and h ?

 

[Inventive]

Hi those derivatives will tell you how the functions are changing, if that helps

 

[John Park]

The answer may depend on what limitations you place on f, g and h in terms of continuity and differentiability. If f is continuous I can almost trivially construct functions g and h that would satisfy your criteria but be only piecewise continuous.

 

[LCKurtz]

A function $f$ from $[a,b]$ to $\mathbb R$ is the difference of two monotonic functions if and only if $f$ is of bounded variation. See, for example, http://www.math.ubc.ca/~feldman/m321/variation.pdf.