# B Sum of increasing and decreasing functions

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1. Mar 10, 2017

### 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

2. Mar 10, 2017

### 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 ?

3. Mar 10, 2017

### Inventive

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

4. Mar 13, 2017

### 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.

5. Mar 13, 2017

### 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.