**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

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$.