  This paper is devoted to the approximation of differentiable semialgebraic
functions by Nash functions. Approximation by Nash functions is known for
semialgebraic functions defined on an affine Nash manifold M, and here we
extend it to functions defined on Nash subsets X of M whose singularities are
monomial. To that end we discuss first "finiteness" and "weak normality" for
such sets X. Namely, we prove that (i) X is the union of finitely many open
subsets, each Nash diffeomorphic to a finite union of coordinate linear
varieties of an affine space and (ii) every function on X which is Nash on
every irreducible component of X extends to a Nash function on M. Then we can
obtain approximation for semialgebraic functions and even for certain
semialgebraic maps on Nash sets with monomial singularities. As a nice
consequence we show that m-dimensional affine Nash manifolds with divisorial
corners which are class k semialgebraically diffeomorphic, for k>m^2, are also
Nash diffeomorphic.
