  Let $(M, g)$ be a closed Riemannian manifold, where g is $C^1$-smooth metric.
Consider the sequence of eigenfunctions $u_k$ of the Laplace operator on M. Let
$B$ be a ball on $M$. We prove a sharp estimate of the number of nodal domains
of $u_k$ that intersect $B$. The problem of local bounds for the volume and for
the number of nodal domains was raised by Donnelly and Fefferman, who also
proposed an idea how one can prove such bounds. We combine their idea with two
ingredients: the recent sharp Remez type inequality for eigenfunctions and the
Landis type growth lemma in narrow domains.
