# algebraic vis-a-vis analytic functions

Consider functions $\mathbb{R}^n\to\mathbb{R}$. If the set of algebraic functions is the algebraic closure of the rationals, does this mean that limits of polynomials are algebraic? In particular, are real-analytic functions algebraic?

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Limits in what sense? –  Thierry Zell Jan 11 '11 at 0:39
Are we overloading the word "closure"? –  Igor Rivin Jan 11 '11 at 0:40
Isn't $f(x)=sin(x)$ a limit of polynomials but not the solution to a real rational runction? –  Sean Tilson Jan 11 '11 at 0:47
Point-wise limits with respect to the standard distance. –  Joakim Jan 11 '11 at 0:47