Remark 1

All the preceding propositions, stated for functions differentiable at a point $x_0$, immediately yield propositions for functions which are right (resp. left) differentiable at $x_0$, when, instead of the functions themselves, one considers their restrictions to the intersection of their intervals of definition with the interval $[x_0, +\infty[$ (resp. ]$-\infty, x_0$]); we leave it to the reader to state them.