We all know that for a differentiable function $f$ on $\mathbb R$,its derivative function can have only essential or $2$nd kind discontinuity.Now my question is if $X$ be that set of all points of discontinuities of the derivative function,then can we predict the nature of $X$,i.e.can we say anything about the structure of $X\subset \mathbb R$.
Asked
Active
Viewed 23 times
0
-
The set of discontinuities is a dense $F_{\sigma}$ set. I don't kn ow if every dense $F_{\sigma}$ set arises this way. – Kavi Rama Murthy Dec 27 '19 at 05:18
-
@KaviRamaMurthy Woulf you explain elaborately? – Kishalay Sarkar Dec 27 '19 at 05:45
-
The set of discontinuity points of any function is an $F_{\sigma}$. Now $f'(x)=\lim \frac {f(x+\frac 1n)-f(x)} {\frac 1n}$ and the point-wise limit of any sequnce of continuous functions is continuous on a dense set. This can be proved using Baire Category Theorem. – Kavi Rama Murthy Dec 27 '19 at 05:48