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typo fix and add extra contents

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Trance-0
2025-02-22 16:35:08 -06:00
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commit 7699fcaea1
6 changed files with 94 additions and 26 deletions
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pages/Math4121/Math4121_L14.md
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@@ -2,7 +2,7 @@
## Recap
### Hankel developed Riemann's integrabilty criterion.
### Hankel developed Riemann's integrability criterion
#### Definition: Oscillation
@@ -22,9 +22,13 @@ $$
where $\mathcal{P}=\{i:\omega(f,I_i)>\sigma\}$.
Proof as homework questions.
Proof:
#### Corollary 2.4
To prove Riemann's Integrability Criterion, we need to show that a bounded function $f$ is Riemann integrable if and only if for every $\sigma, \epsilon > 0$, there exists a partition $P$ of $[a, b]$ such that the sum of the lengths of the intervals where the oscillation exceeds $\sigma$ is less than $\epsilon$.
EOP
#### Proposition 2.4
For point $c\in[a,b]$, define the oscillation at $c$ as
@@ -54,7 +58,7 @@ $$
c_e(S) = \inf_{c\in C_s}\ell(C)
$$
where $\C_s$ is the set of all finite covers of $S$.
where $C_s$ is the set of all finite covers of $S$.
Example: