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The Weierstrass (Polynomial) Approximation TheoremLet be continuous on a real interval . Then for any , there exists an th-order polynomial , where depends on, such that

for all .Thus, any continuous function can be approximated arbitrarily well by means of a polynomial. Furthermore, an infinite-order polynomial can yield an error-free approximation. Of course, to compute the polynomial coefficients using a Taylor series expansion, the function must also be differentiable of all orders throughout .