science forum beginner
Joined: 12 Dec 2005
|Posted: Sat Jul 08, 2006 8:17 am Post subject:
function semi-algebraic over real closed sub-field?
Hi. Getting right to the point:
Let F be a real-closed subfield of the reals R.
Let f : [0,1]^n -> R denote a continuous real
function, semi-algebraic (over R), satisfying that
f(x) lies in F for every rational x (i.e. on Q).
Does this imply f to be semi-algebraic over F?
(It does under the additional hypothesis
that f be a rational function: solve the
interpolation problem for (x,f(x)), x in Q )
If it does not,
how about if f is differentiable or analytic?
Thanks for listening