Definition of total variation: I cannot grasp its meaning

definitionreal-analysistotal-variation

Starting from this definition of Total Variation for a function of one real variable, I cannot understand what it does mean that "supremum runs over set of all partitions".

My interpretation:
I have a function $f:x\in(a,b)\mapsto f(x)$, I can figure out infinitely many partitions $P_{n\in\mathbb{N}}=(x_0^n,\ldots,x_i^n)$ of the interval $(a,b)$ such that:

  • $$P_1=(x_0^1,\ldots,x_i^1) \text{ has a specific sum }J_1=\sum_{j=0}^{i-1}|f(x_{j+1}^1)-f(x_j^1)| \text{ associated to itself} \tag{1}$$
  • $$P_2=(x_0^2,\ldots,x_i^2) \text{ has a specific sum }J_2=\sum_{j=0}^{i-1}|f(x_{j+1}^2)-f(x_j^2)| \text{ associated to itself} \tag{2}$$
    $$\vdots$$

Eventually, I compute total variation of $f(x)$ such that:
$$TV(f(x))=\sup(J_1,J_2,\ldots)\tag{3}$$
where $TV$ stands for Total Variation.

I do not know at all whether my reasoning is correct. If it were so, even if I start from a "larger-mesh" partition $P_1$ and go on with partitions whose mesh is lower than the previous partition mesh, why should I expect that $J_1\ne J_2\ne \ldots$?
I know that maybe my point is very very silly, but I would guess that, whatever the partition is (in terms of mesh length), the sum of the absolute value of step-wise variations of $f$ is always the same, namely that $J_1=J_2=\ldots$
Hence, my question would be: why taking several partitions and then taking supremum over all such different partitions? Woldn't it suffice to take just one single general partition?

Finally, if I had to graphically imagine what total variation of a real function of one variable is, should I imagine it as the sum of step-wise length of corresponding $y$-axis couple of points?

Sorry in advance if my reasoning ignores very trivial facts.

Best Answer

You cannot enumerate the set of partitions of $[a,b]$. In other words, there is no list $\{P_1,P_2,\ldots\}$ which consists of all partitions of $[a,b]$. Other than that, you are correct:

  1. You consider the set $\mathcal P$ of all partitions of $[a,b]$.
  2. For each $P\in\mathcal P$, if $P=\{x_0,x_1,\ldots,x_n\}$, you compute the sum $\displaystyle J_P=\sum_{i=0}^{n-1}\bigl|f(x_i)-f(x_{i+1})\bigr|$.
  3. Then the total variation of $f$ is $\sup\{J_P\mid P\in\mathcal P\}$.

Concerning your question but whether you could take the sum with respect to a “single general partition”, you will have to explain first what that is. Note that, in general, if $P,P^\star\in\mathcal P$, then $J_P\ne J_{P^\star}$. And, (again, in general) there is no partition $P$ such that the total variation is equal to $J_P$; in other words, that supremum is not, in general, a maximum.

Related Question