Suppose that $f \in C^k(\overline U)$, $U \subset \mathbb{R}$ open. Then given a multi-index $|\alpha| \le k$, one has that $D^\alpha f$ can be uniquely extended to $\partial U$. Since this holds for $k =0$, we note that $f$ can be uniquely extended to $\partial U$ as well.
For the sake of notation, assume we're in dimension one. It appears then that we have two possible definitions for $f'$. Fixing $x_0 \in \partial U$, the two logical such definitions are
\begin{align*} f'(x_0) = \lim\limits_{x \rightarrow x_0} \dfrac{f(x)-f(x_0)}{x-x_0} \end{align*}
where $f(x_0)$ is the unique extension of $f$ to $x_0 \in \partial U$, and the limit is taken over $x \in U$, or
\begin{align*} f'(x_0) = \lim\limits_{x \rightarrow x_0} f'(x) \end{align*}
where again, we consider only $x \in U$ (the latter is the definition given in the book, and the first is just a conjecture). I can only prove that these two coincide when the point $x_0$ has the property that some $B(x_0, r) \cap U$ is connected, (i.e., I use the mean value theorem to estimate some term involving a difference quotient and a derivative). Is there a proof for this in general? Are these always the same?