  We prove that, for any closed semialgebraic subset $W$ of $\mathbb{R}^n$ and
for any positive integer $p$, there exists a Nash function
$f:\mathbb{R}^n\setminus W\longrightarrow (0, \infty)$ which is equivalent to
the distance function from $W$ and at the same time it is $\Lambda_p$-regular
in the sense that $|D^\alpha f(x)|\leq C d(x, W)^{1- |\alpha|}$, for each $x\in
\mathbb{R}^n\setminus W$ and each $\alpha\in \mathbb{N}^n$ such that $1\leq
|\alpha|\leq p$, where $C$ is a positive constant. In particular, $f$ is
Lipschitz. Some applications of this result are given.
