I am interested in knowing what we can say in general about when a continuous function $f:\mathbb{R} \to \mathbb{R}$ is differentiable.
To my mind, there are various ways a continuous function can fail to be differentiable. It could have a corner (i.e. its left-derivative is not equal to its right-derivative, but both exist). It could oscillate wildly, like $x\sin \dfrac{1}{x}$ at $x=0$. I'm not really sure if there are other options.
For instance, suppose we eliminate the oscillation by saying that $f$ is monotone on $[a,b]$. Can we then say for instance that its left-derivative exists almost everywhere?
Best Answer
A monotone function is differentiable almost everywhere according to a theorem of Lebesgue. See here for an elementary proof.