The Weierstrass approximation theorem assures us that polynomial
can get arbitrarily close to any continuous function as
the polynomial order is increased.
be continuous on a real interval
. Then for any
, there exists an
, such that
For a proof, see, e.g.
, 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
can be used to find such a polynomial since, in
particular, the function must be differentiable of all orders
. 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
any continuous function.
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