Adjusted mean value theorem.

src/dg/derivative/sets.tex

@@ -143,11 +143,12 @@
     Let $E$ be a separated topological vector space and $\sigma \subset B(\real)$ be an upward-directed system that contains finite sets, then
     \begin{enumerate}
         \item $\mathcal{R}_{\sigma}(\real; E) = \mathcal{R}_{B(\real)}(\real; E)$. Hence, all forms of $\sigma$-differentiability on $\real$ are equivalent.
-        \item For any $U \subset \real$ open, $f: U \to E$, and $x_0 \in U$, $f$ is ($\sigma$-)differentiable at $x_0$ if and only if
+        \item For any $U \subset \real$ open, $f: U \to E$, and $x_0 \in U$, $f$ is differentiable at $x_0$ if and only if
         \[
         \lim_{t \to 0}\frac{f(x + t) - f(x)}{t}
         \]
-        exists. In which case, the above limit is identified with the ($\sigma$-)derivative of $f$ at $0$.
+        exists. In which case, the above limit is identified with the derivative of $f$ at $0$.
+        \item For any $U \subset \real$ open, $f: U \to E$, and $x_0 \in U$, if $f$ is differentiable at $x_0$, then $f$ is continuous at $x_0$.
     \end{enumerate}
 \end{proposition}
 \begin{proof}