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Weierstraß Approximation Theorem

If $f$ is a continuous real-valued function on $[a,b]$ and if any $\epsilon > 0$ is given, then there exists a Polynomial $p$ on $[a,b]$ such that

\vert f(x)-P(x)\vert<\epsilon

for all $x\in[a,b]$. 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.

See also Müntz's Theorem

© 1996-9 Eric W. Weisstein