Manfred Krüppel
On the Zeros of an Infinitely Often Differentiable Function and their Derivatives
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 59, 63-70(2005)

MSC:

45D05 Volterra integral equations, See also {34A12}

39B22 Equations for real functions

34K15 Qualitative theory

26A30 Singular functions, Cantor functions, functions with other special properties

41A15 Spline approximation

Abstract: In this paper, we investigate the structure of
an infinitely often differentiable real function $f$ defined on the
interval $[0,1]$. We show that for such a function the set $\{t: \exists
n\in\mathbb N_0: f^{(n)}(t)=0,\; f^{(n+1)}(t)\not=0\,\}$ is at most
countable, and if $f$ is not a polynomial then the set
$\{t:f^{(n)}(t)\not=0,\; \forall n\in\mathbb N_0\}$ has the power $\bf
c$.
Keywords: $C^\infty$-functions, derivatives of higher order, Cantor sets, Theorem of Cantor-Bendixsohn, sets of first category.