update

pages/Math4121/Math4121_L22.md

@@ -2,8 +2,6 @@
 
 ## Continue on Arzela-Osgood Theorem
 
-
-
 Proof:
 
 Part 2: Control the integral on $\mathcal{U}$
@@ -100,7 +98,7 @@ Part 2: If $f$ is pointwise discontinuous, then $\mathcal{D}$ is of first catego
 
 Let $P_k=\{x\in [a,b]: w(f;x)\geq \frac{1}{k}\}$, $\mathcal{D}=\bigcup_{k=1}^\infty P_k$.
 
-Need to show that each $P_k$ is nowhere dense. (under the assumption that $\mathcal{C)$ is dense).
+Need to show that each $P_k$ is nowhere dense. (under the assumption that $\mathcal{C}$ is dense).
 
 Let $I\subseteq [a,b]$ so $\exists c\in \mathcal{C}\cap I$. So by definition of $w(f;c)$, $\exists J\subseteq I$ and $c\in J$ such that $w(f;J)\leq \frac{1}{k}$ so for all $x\in J$, $w(f;x)\leq \frac{1}{k}$. so $J\subseteq P_k=\emptyset$.