  We show that differentiable functions, defined on a convex body $K \subseteq
\mathbb R^d$, whose derivatives do not exceed a suitable given sequence of
positive real numbers share many properties with polynomials. The role of the
degree of a polynomial is hereby played by an integer associated with the given
sequence of reals, the diameter of $K$, and a real parameter linked to the
$C^0$-norm of the function. We give quantitative information on the size of the
zero set, show that it admits a local parameterization by Sobolev functions,
and prove an inequality of Remez-type. From the latter, we deduce several
consequences, for instance, a bound on the volume of sublevel sets and a
comparison of $L^p$-norms reversing H\"older's inequality. The validity of many
of the results only depends on the derivatives up to some finite order; the
order can be specified in terms of the given data.
