## Weierstrass Approximation Theorem

The Weierstrass approximation theorem assures us that polynomial approximation can get arbitrarily close to any continuous function as the polynomial order is increased. Let be continuous on a real interval . Then for any , there exists an th-order polynomial , where depends on , such that

*e.g.*, [63, pp. 146-148]. Thus, any continuous function can be approximated arbitrarily well by means of a polynomial. This does not necessarily mean that a Taylor series expansion can be used to find such a polynomial since, in particular, the function must be differentiable of all orders throughout . Furthermore, there can be points, even in infinitely differentiable functions, about which a Taylor expansion will not yield a good approximation, as illustrated in the next section. The main point here is that, thanks to the Weierstrass approximation theorem, we know that good polynomial approximations

*exist*for any continuous function.

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