Added remark reflecting on past mistakes.

src/fa/rs/path.tex

@@ -52,7 +52,7 @@
 \begin{proof}
     Let $[\cdot]_E: E \to [0, \infty)$ and $[\cdot]_F: F \to [0, \infty)$ such that for any $x \in E$ and $y \in F$, $[xy]_G \le [x]_E[y]_F$. Since $\gamma([a, b])$ is compact, by modifying $[\cdot]_F$, assume without loss of generality that there exists $V \in \cn_F(\gamma([a, b]))$ such that for any $x, y \in V$ with $[x - y]_F \le 1$, $[f(x) - f(y)]_E \le \eps$.
 
-    Since $f \in C(U; E)$, $f \in PI([a, b], \gamma; E)$ by \autoref{proposition:rs-bv-continuous}. Given that $\gamma$ is of bounded variation, there exists $(P  = \seqfz{x_j}, c) \in \scp_t([a, b])$ such that:
+    Since $f \in C(U; E)$, $f \in PI([a, b], \gamma; E)$ by \autoref{proposition:rs-bv-continuous}. Given that $\gamma$ is continuous, there exists $(P  = \seqfz{x_j}, c) \in \scp_t([a, b])$ such that:
     \begin{enumerate}[label=(\alph*)]
         \item For each $1 \le j \le n$, 
         \[
@@ -74,6 +74,12 @@
     
 \end{proof}
 
+\begin{remark}
+\label{remark:piecewise-linear-remark}
+    Past me made the mistake of believing that in \autoref{lemma:rectifiable-piecewise-linear}, it is possible to approximate rectifiable curves with piecewise linear curves in \textit{total variation distance}. However, this is not possible: as every piecewise linear curve is absolutely continuous, and the limit of these curves in total variation distance must also be absolutely continuous. As such, this strong approximation exists if and only if the curve is absolutely continuous. 
+\end{remark}
+
+
 
 \begin{theorem}[Fundamental Theorem of Calculus for Path Integrals]
 \label{theorem:ftc-path-integrals}