
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
for all

.
For a proof, see,
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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