What is the name of this theorem? I do not know how to type some symbols in google. In my textbook, the following theorem is proved.
But I do not understand it. So I am finding some other documents make me understand the theorem. Could someone let me know this theorem's name? Since I don't know the name, it is too difficult to find this theorem.

$f$ is a real-valued function on $[a, b]$.
$\Gamma$ is a partition of $f$ on $[a, b]$.
$$
\Gamma=\left\{x_i\right\}_{i=0}^N, \quad x_0(=a) \lt x_1 \lt x_2 \lt \cdots \lt x_N(=b)
$$
$|\Gamma|$ is a norm of the partition $\Gamma$.
$$
|\Gamma|=\max_{i\in\left\{1, 2, \cdots, N\right\}}\left\{x_i-x_{i-1}\right\}
$$
$S_\Gamma$ is a sum of the absolute of the differences of $f$.
$$
S_\Gamma=\sum_{i=1}^{N}\left|f(x_i) - f(x_{i-1})\right|
$$
$V$ is variance of $f$.
$$
V=\sup_{\Gamma}S_\Gamma
$$

Theorem
If $f$ is continuous on $[a, b]$ then $\displaystyle V=\lim_{\left|\Gamma\right|\to0}S_\Gamma$.
That is, given $M$ satisfying $M<V$, there exists $\delta > 0$ such that $\displaystyle S_\Gamma>M$ with $\left|\Gamma\right|<\delta$.

Actually, I don't know why this theorem is referred in textbook.
In my opinion, $\displaystyle V=\lim_{\left|\Gamma\right|\to0}S_\Gamma$ is sure and trivial for the following two reasons:


*

*The more finely the partition is split, the larger $\displaystyle S_\Gamma$ is; that is, $S_\Gamma \uparrow$ as $|\Gamma|\to0$

*$\displaystyle V=\sup_\Gamma S_\Gamma$.



 A: If what you said were true the theorem would indeed be obvious. But it's  not true that $S_\Gamma$ increases as $|\Gamma|$ decreases.
Say $[a,b]=[0,5]$. Say $f(0)=f(1)=f(2)=f(3)=f(5)=0$, $f(4)=1$. Consider the two partitions $\Gamma_1 = (0,4,5)$, $\Gamma_2=(0,1,2,3,5)$. Then $|\Gamma_2|<|\Gamma_1|$ but $S_{\Gamma_2}<S_{\Gamma_1}$.
Edit: Of course it is true that if $\Gamma'$ is a refinement of $\Gamma$ then $S_{\Gamma'}\ge S_\Gamma$. That's crucial in proving the theorem:
Say $f$ is continuous on $[a,b]$ and $M<V$. Choose a partition $\Gamma_1$ with $$S_{\Gamma_1}>M.$$ Say $\Gamma_1$ has $N$ points (including the endpoints).
Let $\epsilon>0$. Choose $\delta>0$ so that $|s-t|<\delta$ implies $|f(s)-f(t)|<\epsilon/(3N)$. Say $\Gamma_2$ is a partition with $|\Gamma_2|<\delta$. Let $\Gamma$ be the partition obtained by using all the points in $\Gamma_1$ and all the points in $\Gamma_2$. Then $\Gamma$ is a refinement of $\Gamma_1$, so $$S_{\Gamma}\ge S_{\Gamma_1}>M.$$
But $S_{\Gamma_2}$ is the same as $S_\Gamma$, except for $N$ places where $S_{\Gamma}$ includes two terms $$|f(x_j)-f(y_k)|+|f(y_k)-f(x_{j+1})|$$and $S_{\Gamma_1}$ includes just the one term $$|f(x_j)-f(x_{j+1})|$$
Each of those "error terms" is less than $\epsilon/(3N)$. So
$$|S_{\Gamma_2}-S_\Gamma|<\epsilon,$$hence $$S_{\Gamma_2}>M-\epsilon.$$
