Why continuity at a point one of Dini derivatives implies the continuity at this point others Dini derivatives? Let $f:[a,b] \rightarrow \mathbb R$ be a continuous function and $x_0\in (a,b)$. How to  prove that if the Dini derivatives $D^+f(x_0)$ is finite and continuous at $x_0$ then also $D_+f(x_0)$ is continuous at $x_0$.
Thanks
 A: I give you another reference $\;$ Boas - A Primer of Real Functions - (1981) $\,$ pp. 142-143 $\,$ where you can read what you need:

if $f$ is continuous, all four Dini derivatives have the same upper and lower bounds in any interval

and find the proof.
Now suppose, for instance, that $D^+f$ is finite and continuous at $x_0$ and take $\varepsilon>0$.
Then there is a $\delta>0$ so that $$D^+f(x_0)-\varepsilon<D^+f(x)<D^+f(x_0)+\varepsilon$$ for $x \in [a,b]$ and $$x \in \;]x_0-\delta,x_0+\delta[$$ The same happens for $D_+f$ for the above-mentioned reason, whence the continuity of $D_+f$ in $x_0$ .
If you are interested, Dini's original argument (in Italian of course) is contained in his Fondamenti per la teorica delle funzioni di variabili reali (1878) on pp. 194-195 of UMI edition (1990) :

... gli estremi oscillatorii inferiori e superiori destro e sinistro $\lambda_x\,$,$\Lambda_x\,$,$\lambda_x'\,$,$\Lambda_x'\,$ hanno tutti e quattro uno stesso limite inferiore, e uno stesso limite superiore in qualsiasi porzione dell'intervallo nella quale si considerino ...

