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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.

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  • $\begingroup$ If $f$ has a positive derivative, you can rewrite it as the sum of an increasing and a constant function. $\endgroup$ – Gabriel Romon Mar 12 '14 at 7:50
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I assume you mean "non decreasing" instead of increasing, and "non increasing" instead of decreasing. Let $g$ and $h$ be the nondecreasing and nonincreasing summands, respectively. Note that we will define $g$ and $h$ so that they are continuous. Let $g(a) = 0$ and $h(a) = f(a)$. You can simply make $g$ constant whenever the derivative of $f$ is nonpositive and $h$ constant whenever it is nonnegative. Suppose $f$ has a local maximum at $c$ and its next local minimum at $d$, so it is decreasing on $(c, d)$: then on $(c, d)$, $h(x) = f(x) - r$, where $r = g(c)$.You can similarly define $g$'s graph to be a translated copy of the graph of $f$ on any interval where $f$ is increasing.

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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 $\int_a^x g_2(t)\; dt$.

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