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 $f(x)$ be continuous on a real interval $I$. Then for any $\epsilon>0$, there exists an $n$th-order polynomial $P_n(f,x)$, where $n$ depends on $\epsilon$, such that

$$\left\vert P_n(f,x) - f(x)\right\vert < \epsilon$$

for all $x\in I$.

For a proof, see, e.g., [51, 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 $I$. 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.