Let $f : [a,b] \rightarrow \mathbb{R}$ be a $C^1$ function such that $f$ is monotonic on each $[t_k, t_{k+1}]$, with $a = t_0 < t_1 < ... < t_N = b$.

Let g be the increasing-ified version of $f$, i.e. on each interval where $f$ is decreasing we define $g(x) = -f(x) + constant$, such that the function $g$ is continuous.

More precisely :

  • if $f$ is increasing or constant on $[t_0, t_{1}]$, then $g = f$ on this interval

  • if $f$ is decreasing on $[t_0, t_{1}]$, then $g = -f$ on this interval and thus $g$ is increasing on this interval

  • we do the same on each following interval $[t_k, t_{k+1}]$ : if $f$ is decreasing, we set $g(x) = -f(x) + \alpha_k$, where $\alpha_k$ is chosen such that $g$ is continuous.

Example : $f(x) = \sin(x)$ in red, the function $g$ in green:


1) This concept surely exists somewhere. How is it called?

2) Without loss of generaly, let's assume $a=0$ and $f(0)=0$. It seems that $g$ is :

$$ g(x) = \int_0^x | f'(t)| d t.$$

Is that true?

3) It seems that $R(x) = g(x) / x$ looks like a good measure of how much $f(t)$ "moves" vertically when $t$ goes from $0$ to $x$, i.e. :

  • if $g(x) / x$ is close to zero, $f$ has very little variation (nearly constant) on $[0, x]$

  • if $g(x) / x$ is big, $f$ has much variation on $[0, x]$

Does this ratio have a name?

Example: with the previous example, $R(10) \simeq 6.54 / 10 = 0.654$

Example: with $f(x) = \sin(x^2)$, we have $R(10) \simeq 63.49 / 10 = 6.349$

Note: now having written this whole thing, I thing this is related to length of arc length / rectification. But still, I'd like to know more about these things.

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    $\begingroup$ $g$ is the "total variation" of the function $f$. $\endgroup$ – Martin R Dec 18 '15 at 0:01
  • 1
    $\begingroup$ Very cool question. $\endgroup$ – goblin Dec 18 '15 at 1:05

To summarize what Martin R pointed out:

  1. $g$ is the total variation function, it can be written as $g(x) =V_a^x f$ where $V_a^x$ is the total variation of $f$ restricted to $[a,x]$.

  2. Yes, for $C^1$ smooth functions $V_a^x f= \int_a^x |f'(t)|\,dt$. The same holds more generally: whenever $f$ has finite variation on $[a,x]$, it is differentiable almost everywhere, and the integral of $|f'|$ gives the variation.

  3. The ratio $\frac{1}{x-a}V_a^x f$ does not appear to have a name; it could be called mean variation of $f$ on $[a,x]$, or the running average of $|f'|$. Its distant relative is mean oscillation.

Note: For a real-valued continuous function f, defined on an interval [a, b] ⊂ ℝ, its total variation on [a, b] is a measure of the one-dimensional arclength of the curve with parametric equation x ↦ f(x), for x ∈ [a, b].


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