If $f\in C^1[a,b]$, then it can be expressed by the sum of an increasing function and a decreasing function. Prove: If $f$ is a continuous class $1$ function on $[a,b]$ then it can be expressed by the sum of an increasing function and a decreasing function.
I don´t know where to start my demonstration, I really would appreciate your help please.

Let $g_1 = \max(f', 0)+1$ and $g_2 = \min(f',0) - 1$. 
Consider $f(a) + \int_a^x g_1(t)\; dt$ and $\int_a^x g_2(t)\; dt$.
