Polynomial density and Stone–Weierstrass

Weierstrass approximation says that every continuous real function on a finite closed interval is the uniform limit of polynomials. A sequence of polynomials may therefore approach the absolute-value function despite its corner at zero. The approximating derivatives need not approach a continuous derivative, and no contradiction arises with the earlier derivative-limit theorem. Uniform approximation controls values; it does not promise smooth convergence or a finite polynomial equal to the target.

Nor must the approximants be Taylor partial sums. A function may fail to be differentiable or may be smooth without equaling its Taylor series. Density allows coefficients to depend on global information and on the desired tolerance. This distinction matters when choosing numerical representations: approximation existence does not automatically give an efficient degree, a stable coefficient basis, or resistance to noisy samples. Those are quantitative and computational questions beyond the qualitative density theorem.

For compact Hausdorff K, the real Stone–Weierstrass theorem applies to a subalgebra of continuous real-valued functions containing the constants and separating points. Subalgebra means closure under finite sums, scalar multiplication, and products. Separation means that for any distinct locations, at least one family member takes different values there. Under these assumptions, the algebra is dense in all continuous real functions with the supremum norm. The compact interval polynomial theorem is the case generated by the coordinate function.

Each structural requirement removes a real obstruction. Polynomials lacking a constant term cannot uniformly approximate the constant one on an interval containing zero. Even polynomials cannot approximate x on a symmetric interval, because they never distinguish x from minus x. Merely taking affine functions supplies constants and separates points but fails product closure, and it cannot approximate arbitrary curved profiles arbitrarily well. For complex-valued algebras one needs the additional conjugation closure in the standard complex theorem; the real statement cannot be copied unchanged.

On the unit interval, form a polynomial by weighting sampled values f(k/n) with binomial probabilities. At input x this is the expectation of f(K/n) when K has binomial parameters n and x. The sample fraction has mean x and variance x(1−x)/n, at most one over four n. Continuity on the compact interval is uniform continuity, so nearby sample fractions produce nearby function values with one location-independent tolerance.

Split the expectation into a nearby region and its complement. Uniform continuity bounds the nearby error; boundedness of f bounds the far error, while Chebyshev’s inequality makes the far probability uniformly small using the variance estimate. This proves uniform approximation without assuming differentiability. In the binomial lab, p plays the role of the approximation input and n the polynomial degree: predict how the distribution of K/n concentrates as n increases. The bars illustrate the weights, not an arbitrary target function or an automatic polynomial fitter.