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 $$ |f(x)-p(x)|<\epsilon $$ 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.

Convergence of sequences of functions is a major topic in Analysis. The Weierstrass approximation theorem states that the restrictions to $[a,b]$ of the polynomial functions are dense in $\mathcal{C}\bigl([a,b]\bigr)$ with respect to the supremum norm. In other words, every $f\in\mathcal{C}\bigl([a,b]\bigr)$ is the uniform limit of a sequence of polynomial functions.

A generalization of this theorem in the context of continuous functions defined on compact topological spaces is the Stone-Weierstrass theorem

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