Typical bad differentiable extensions (Q2414837)

An almost century old result by \textit{V. Jarník} [Rozpravy 32, Nr. 15, 15 S. (1923; JFM 49.0705.01)] says that any differentiable function $\varphi: P\to\mathbb R$, $P$ a closed subset of $\mathbb R$, has a differentiable extension onto $\mathbb R$. In their previous paper on this topic [J. Math. Anal. Appl. 464, No. 1, 274--279 (2018; Zbl 1388.26008)], the authors have shown that Jarník's extension can be nowhere monotone in $\mathbb R\setminus P$. The present item arrives as a follow-up to that work. With the metric $\varrho(f,g)=\min{\{1,\|f-g\|_{C^1}\}}$, $\|f\|_{C^1}=\|f\|_\infty+\|f'\|_\infty$, considered in the space $E_\varphi(\mathbb R)$ of all differentiable extensions of some fixed $\varphi$, it is shown that extensions being nowhere monotone in $\mathbb R\setminus P$ form a $G_\delta$ dense (so, generic) subset of $E_\varphi(\mathbb R)$. On the other hand, in a less obvious result, the authors prove that extensions being monotone somewhere in $\mathbb R\setminus P$, nevertheless, form a dense subset in $E_\varphi(\mathbb R)$. The basic tool used in the proofs is the classical \textit{D. Pompéiu}'s construction of an increasing differentiable function $h:[0,1]\to\mathbb R$ having the set of saddle points ($h'(x)=0$) metrically dense in $[0,1]$ [Math. Ann. 63, 326--332 (1907; JFM 38.0425.04)].