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Consider a bounded function $f:[0,1]\to\Bbb R$.

  1. Can $f$ be written as the sum of a non-decreasing and non-increasing function?

  2. What if $f \in C^1[0, 1]$? Can it be written as the sum of a non-decreasing and non-increasing function?

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Every monotone function is continuous except at most countably many points. Thus a function that is discontinuous everywhere on $[0, 1]$ cannot be written as the sum of a non-decreasing function and a non-increasing function.

However, functions in $C^1[0,1]$ do have the desired decomposition. To see this, suppose $f \in C^1[0,1]$. Define functions $g$ and $h$ on $[0, 1]$ as follows: $$ g(x) = \begin{cases} f'(x) &\text{ if } f'(x) \ge 0 \\ 0&\text{ if } f'(x) < 0 \end{cases} \quad \text{and} \quad h(x) = \begin{cases} 0 &\text{ if } f'(x) \ge 0 \\ f'(x)&\text{ if } f'(x) < 0. \end{cases} $$ Then $f'(x) = g(x) + h(x)$ for every $x \in [0, 1]$. Hence $$ f(x) = f(0) + \int_0^x g + \int_0^x h $$ for every $x \in [0, 1]$. Note that $f(0) + \int_0^x g$ is a non-decreasing function of $x$ and $\int_0^x h$ is a non-increasing function of $x$. Thus we have the desired decomposition.

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  • $\begingroup$ Perhaps, mention that the functions $g$ and $h$ defined here are also continuous (since $f'$ is). $\endgroup$ – B. S. Thomson Dec 1 '15 at 1:54
  • $\begingroup$ What do you mean by "Every monotone function is continuous except at most countably many points." Thanks for your help. $\endgroup$ – Spencer Ireland Dec 1 '15 at 2:59
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    $\begingroup$ A function is called monotone if it is either non-decreasing or non-increasing. $\endgroup$ – Sheldon Axler Dec 1 '15 at 3:19

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