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We define Schwartz functions and tempered functions on
affine real algebraic varieties, which might be singular. We prove
that some of the important classical properties of these functions,
such as partition of unity, characterization on open subsets, etc.,
continue to hold in this case. The study of this category (joint
with Ary Shaviv) was later enlarged to include Nash manifolds. The
new category was named Quasi Nash and indeed it contains the Nash
category as full subcategory, and the algebraic case as (non full)
subcategory. Furthermore, the new category enables us to prove some
non-affine properties which we were unable to show in the general
algebraic case.
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