Typical bad differentiable extensions.

Abstract By a 1923 result of V. Jarník, every differentiable map φ from a closed subset P of R into R has a differentiable extension f : R → R. It has been recently proved, by the authors, that among such differentiable extensions of φ there is always one that is nowhere monotone on R ∖ P. In particular, the family E φ 1 (R) of "bad" differentiable extensions f : R → R of φ , for which the set [ f ′ = 0 ] : = { x ∈ R : f ′ (x) = 0 } is dense in R ∖ P , is nonempty. We notice here that E φ 1 (R) with a natural distance is a complete metric space and prove that actually a typical function in E φ 1 (R) is nowhere monotone on R ∖ P. At the same time, the set M φ (R) , of functions f ∈ E φ 1 (R) which are monotone on some nonempty subinterval of every nonempty open U ⊂ R ∖ P , is dense in E φ 1 (R). This last statement remains true, when the term "monotone" is replaced with either of the following three terms: "strictly increasing," "strictly decreasing," or "constant." [ABSTRACT FROM AUTHOR]