Weierstrass approximation theorem

Polynomials are the simplest functions we can compute by hand: just additions and multiplications. The Weierstrass approximation theorem makes a remarkable promise — every continuous function on a closed bounded interval, no matter how wiggly or jagged, can be matched as closely as you like, everywhere at once, by a polynomial. Continuity alone is enough; no smoothness required.

Statement: if f is continuous on a closed interval [a, b], then for every epsilon > 0 there is a polynomial p such that |f(x) - p(x)| < epsilon for all x in [a, b]. Equivalently, the polynomials are dense in the space C([a, b]) of continuous functions under the sup norm (the norm of uniform closeness). The approximation is uniform: a single polynomial works across the entire interval, not point by point.

Note what this does NOT say. It does not give a single power series converging to f — the approximating polynomials change as epsilon shrinks and need not be partial sums of one series. Indeed f need not be differentiable anywhere, so it may have no Taylor series at all. A concrete construction uses the Bernstein polynomials, which converge uniformly to f and give an explicit, probabilistic proof.