It is well known that if $f$ is monotone on $[a,b]$, then $f$ is differentiable almost everywhere on $[a,b]$. I am trying to find a condition which forces $f$ to be differentiable at its endpoints (right-differentiable at $a$, say): the weaker the restriction the better. Worst case scenario is the condition being exactly the desired property.
I thought of uniform continuity, bounded derivative, but these all fail. Is there a property I can add to $f$ that is weaker than "right-differentiable at $a$", but that implies the latter?
(would also appreciate a reference if there is such a result - I am trying to incorporate this into a study I am doing)
If $f:[a,b]\to \mathbb R$ is nondecreasing then the right-hand derivative $f'_+(a)$ exists if and only if the approximate right-hand derivative at $a$ exists. Weaker still, this is true if and only if there is a set $E$ that is nonporous on the right at $a$, so that $$\lim_{y\to a+,\, y\in E} \frac{f(y)-f(a)}{y-a}$$ exists. If you have a right-hand derivative relative to at least one set that is nonporous at $a$ then you have a derivative $f'_+(a)$ .
But this says that to be sure a derivative exists, it is enough if some much weaker derivative exists. Maybe not at all what you were hoping for. If you do like it I can supply the references.