Given $a,b\in\mathbb R$ with $a<b$. Let $$X=\{f\in C([a,b]): f \text{ is differentiable on } [a,b] \text{ with }f' \text{ bounded }\},$$ and $$A=\{f\in X: f' \text{ is Riemann integrable}\}. $$ It is known that $A\subsetneq X$, see this post.

I wonder, is there any result concerning the question that what proportion of $A$ is in $X$? I mean, I'm looking for results similar to "A generic continuous function on $[0,1]$ is nowhere differentiable". Does a "generic" element of $X$ lie in $A$?

If we want to use the Baire category theorem, maybe we need to be careful to define an appropriate metric on $X$ to make it become a complete metric space (but I don't know how to find this appropriate metric).