  We prove a quantitative version of the curve selection lemma. Denoting by
$s,d,k$ a bound on the number, the degree and the number of variables of the
polynomials describing a semi-algebraic set $S$ and a point $x$ in $\bar S$, we
find a semi-algebraic path starting at $x$ and entering in $S$ with a
description of degree $(O(d)^{3k+3},O(d)^{k})$ (using a precise definition of
the description of a semi-algebraic path and its degree given in the paper). As
a consequence, we prove that there exists a semi-algebraic path starting at $x$
and entering in $S$, such that the degree of the Zariski closure of the image
of this path is bounded by $O(d)^{4k+3}$, improving a result of Jelonek and
Kurdyka. We also give an algorithm for describing the real isolated points of
$S$ whose complexity is bounded by $s^{2 k+1}d^{O(k)}$ improving a result of
Le, Safey el Din, and de Wolff.
