  We study robust properties of zero sets of continuous maps
$f:X\to\mathbb{R}^n$. Formally, we analyze the family
$Z_r(f)=\{g^{-1}(0):\,\,\|g-f\|<r\}$ of all zero sets of all continuous maps
$g$ closer to $f$ than $r$ in the max-norm. The fundamental geometric property
of $Z_r(f)$ is that all its zero sets lie outside of $A:=\{x:\,|f(x)|\ge r\}$.
We claim that once the space $A$ is fixed, $Z_r(f)$ is \emph{fully} determined
by an element of a so-called cohomotopy group which---by a recent result---is
computable whenever the dimension of $X$ is at most $2n-3$. More explicitly,
the element is a homotopy class of a map from $A$ or $X/A$ into a sphere.
  By considering all $r>0$ simultaneously, the pointed cohomotopy groups form a
persistence module---a structure leading to the persistence diagrams as in the
case of \emph{persistent homology} or \emph{well groups}. Eventually, we get a
descriptor of persistent robust properties of zero sets that has better
descriptive power (Theorem A) and better computability status (Theorem B) than
the established well diagrams. Moreover, if we endow every point of each zero
set with gradients of the perturbation, the robust description of the zero sets
by elements of cohomotopy groups is in some sense the best possible (Theorem
C).
