  Let $f: B^n \rightarrow {\mathbb R}$ be a $d+1$ times continuously
differentiable function on the unit ball $B^n$, with $\max_{z\in B^n} \Vert
f(z) \Vert=1$. A well-known fact is that if $f$ vanishes on a set $Z\subset
B^n$ with a non-empty interior, then for each $k=1,\ldots,d+1$ the norm of the
$k$-th derivative $||f^{(k)}||$ is at least $M=M(n,k)>0$.
  \medskip
  We show that this fact remains valid for all ``sufficiently dense'' sets $Z$
(including finite ones). The density of $Z$ is measured via the behavior of the
covering numbers of $Z$. In particular, the bound $||f^{(k)}||\ge \tilde
M=\tilde M(n,k)>0$ holds for each $Z$ with the box (or Minkowski, or entropy)
dimension $\dim_e(Z)$ greater than $n-\frac{1}{k}$.

Classical Analysis and ODEs