1

Lemma 1 There is a sequence of partitions $(\Gamma_k)$, where each $\Gamma_{k+1}$ is a refinement of $\Gamma_k$, such that $$ \lim_{k\to\infty}S_{\Gamma_k}=V[f;a,b]. $$

Here there is a similar statement, we call it Lemma 2, without the word refinement. We can prove Lemma 1 by using Lemma 2 to get $(\Gamma_k)$ and define $\Gamma'_{k+1}=\bigcup_{j=1}^k\Gamma_j$.

Is the following "Lemma 3" also valid?

Lemma 3 If $P_n$ is a sequence of partition obtained by bisecting the precedent, does the following statement hold (for a not necessarily continuous map)? How to prove it? $$ \lim_{k\to\infty}S_{P_k}=V[f;a,b]. $$

Notations

  • $S_{\Gamma_k}$ is the variation on the partition $\Gamma_k$
  • $V[f;a,b]$ is the total variation
Community
  • 1
Mohamed Ali
  • 730

1 Answers1

1

If $f$ is not continuous, then Lemma 3 does not always hold.

For a counterexample, take $[a,b] = [0,1]$, let $P_1$ be a uniform partition and let $f$ be a function that assumes a constant value at every rational point but assumes different values at irrational points so that $V_0^1(f) > 0$. In this case , $0 =S_{P_k} \not\to V_0^1(f)$

If $f$ is continuous, then $S_{P_k} \to V_a^b(f)$ when $\|P_k\| \to 0$ as $k \to \infty$.

RRL
  • 90,707