update

pages/Math4111/Math4111_L24.md

@@ -133,6 +133,28 @@ By Theorem 2.28, $\sup f(X)$ and $\inf f(X)$ exist and are in $f(X)$. Let $p_0\i
 
 EOP
 
+---
+
+Supplemental materials:
+
+_I found this section is not covered in the lecture but is used in later chapters._
+
+#### Definition 4.18
+
+Let $f$ be a mapping of a metric space $X$ into a metric space $Y$. $f$ is **uniformly continuous** on $X$ if $\forall \epsilon > 0$, $\exists \delta > 0$ such that $\forall x, y\in X$, $|x-y| < \delta \implies |f(x)-f(y)| < \epsilon$.
+
+#### Theorem 4.19
+
+If $f$ is a continuous mapping of a compact metric space $X$ into a metric space $Y$, then $f$ is uniformly continuous on $X$.
+
+Proof:
+
+See the textbook.
+
+EOP
+
+---
+
 ### Continuity and connectedness
 
 > **Definition 2.45**: Let $X$ be a metric space. $A,B\subset X$ are **separated** if $\overline{A}\cap B = \phi$ and $\overline{B}\cap A = \phi$.