If
is a continuous real-valued function on and if any is given, then there exists a polynomial
on
such that

for all . In words, any continuous function on a closed and
bounded interval can be uniformly approximated on that interval by polynomials
to any degree of accuracy.

Jeffreys, H. and Jeffreys, B. S. "Weierstrass's Theorem on Approximation by Polynomials" and "Extension of Weierstrass's
Approximation Theory." §14.08-14.081 in Methods
of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University
Press, pp. 446-448, 1988.