Proposition 6

*Let $I$ be an open interval in $\mathbf{R}$, let $x_0$ be one of the endpoints of $I$, and $f$ a vector function defined and continuous on $I$, with values in a complete normed space $E$ over $\mathbf{R}$; suppose that $f$ has a right derivative at the points of the complement $B$ in $I$ of a countable subset of $I$. Then for $f'_d(x)$ to have a limit as $x$ tends to $x_0$ while remaining in $B$ and $\neq x_0$ it is necessary and sufficient that $\frac{f(y) - f(x)}{y - x}$ have a limit $c$ as $(x, y)$ tends to $(x_0, x_0)$ subject to $x \in I, y \in I, x \neq x_0, y \neq x_0$ and $x \neq y$. Under these conditions $f$ extends by continuity to the point $x_0$, the right derivative $f'_d(x)$ tends to $c$ as $x$ tends to $x_0$ (while remaining in B) and the function $f$ extended (defined on $I \cup \{x_0\}$) has derivative at $x_0$ equal to $c$.

Suppose for example that $x_0$ is the right-hand endpoint of I. Let us first show that if $f'_d(x)$ tends to $c$ as $x$ tends to $x_0$ while remaining in B and $\neq x_0$, then $\frac{f(y) - f(x)}{y - x}$ tends to $c$; this follows immediately from th. 2 applied to the function $f(z) - cz$, which yields

for $x < y < x_0$. Conversely, if $\frac{f(y) - f(x)}{y - x}$ tends to $c$, then for every $\varepsilon > 0$ there exists an $h > 0$ such that the conditions $|x - x_0| < h,\ |y - x_0| < h\ (x \neq x_0,\ y \neq x_0)$ imply

But for all $x \in B$ and $\neq x_0$ such that $|x - x_0| < h$ there exists a $k > 0$ (depending on $x$) such that the relation $x < y < x + k$ entails

from which, considering (11):

for $|x - x_0| < h,\ x \in B$ and $x \neq x_0$. which proves that $f'_d(x)$ tends to $c$. Moreover, from the relation (11) one has immediately that

which proves (by Cauchy's criterion) that $f$ has a limit $d$ at the point $x_0$ as $x$ tends to this point while remaining in I and $\neq x_0$; now, letting $x$ approach $x_0$ in (11), for $y \in I,\ y \neq x_0$ and $|y - x_0| \leq h$, we have

which proves that $c$ is the derivative at the point $x_0$ of the function $f$ extended by continuity to $I \cup \{x_0\}$.