On the Derivatives of Functions of Bounded Variation

Abstract

The following two questions were submitted by F. S. Cater. Let $\mathcal {F}$ denote the family of all absolutely continuous , nondecreasing functions on $[0,1]$. Endow $\mathcal{F}$ withe the complete metric $d$ defined by $$ d(f,g) = |f(0)-g(0)| + \int_o^1 |f'-g'|$$

Let

$\mathcal{G}=$ {$g\in \mathcal{F}; g'(x) = \infty$ for uncountablely many $x$ in each subinterval of $[0,1]$}.

It is easy to prove that $\mathcal {G}$ and $\mathcal {F}\setminus \mathcal{G}$ are dense subsets of $\mathcal {F}$.

1. Is $\mathcal {G}$ is a first category subset of $\mathcal {F}$?

2. Is $\mathcal {F}\setminus \mathcal {G}$ a first category subset of $\mathcal {F}$?