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Assuming a nicely-behaved function (continuous, differentiable). Total variation can be defined as $T_f=\int_a^b|f'(x)|dx$.

My intuitive understanding of what total variation measures is: the length of a string used to plot that function.

Consider the following functions between the interval $[0,1]$: $f(x)=x,g(x)=x^2$. Considering the fact that the shortest distance between two points is a straight line, I would think $T_f<T_g$. However, performing the calculations yields:

$$ T_f=\int_0^1|1|dx = \int_0^11dx=1 $$ $$ T_g = \int_0^1|2x|dx =\int_0^1 2x=1$$

So, where am I making a mistake? or is my interpretation of total variation wrong; if so, what is a correct intuitive interpretation?

*Edit: I reopened the question because there are answers that address it more directly. **Edit: I can't reopen the question.

Schach21
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    It’s not the length of the string used to plot that function, there’s a different formula for that. It’s the total vertical distance traveled (either moving upward or moving downward) as you walk along the string. If $f$ is the temperature at time $t$ and first the temperature goes up by 20 degrees then the temperature goes down by 10 degrees then the total variation is 30 degrees. – littleO Aug 09 '21 at 01:00
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    I actually don't think the linked duplicate answers your misunderstanding. You say "the length of a string used to plot that function", which I'd reword it to "length of a string used to trace the image of that function". So you're mixing up the roles of the image of the function vs its graph. – peek-a-boo Aug 09 '21 at 01:12
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    Consider $\gamma_f(t)=(t,f(t))$ and $\gamma_g(t)=(t,g(t))$, then you'll see that \begin{align}\int_0^1|\gamma_f'(t)|,dt=\int_0^1\sqrt{1+|f'(t)|^2},dt=\text{length of graph of $f$}=\sqrt{2}\end{align} is strictly smaller than \begin{align}\int_0^1|\gamma_g'(t)|,dt=\int_0^1\sqrt{1+|g'(t)|^2},dt=\text{length of graph of $g$}\approx 1.478.\end{align} – peek-a-boo Aug 09 '21 at 01:12
  • @peek-a-boo thanks! your answer addresses my confusion :) – Schach21 Aug 09 '21 at 01:17

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