Stone-Weierstrass: Examples By Stone-Weierstrass one has: $f\in\mathcal{C}(K):\quad p_n\to f$
Now, for analytic functions this is just Taylor:
$$f\in\mathcal{C}^\omega([a,b]):\quad f(x)=\sum_{k=0}^\infty a_kx^k$$
But, how does this work for other examples like:
$$K:=[-1,1]:\quad f(x):=|x|$$
$$L:=[-1,-\frac12]\cup[\frac12,1]:\quad g(x):=\frac{1}{x}$$
(Note that the latter is over a disconnected space.)


how does this work for other examples  

Not via a Taylor series, that's for sure. The difference between


*

*$f$ is the limit of a uniformly converging power series, 


and 


*$f$ is the limit of a uniformly converging sequence of polynomials 


is that partial sums of a power series are a very special kind of a sequence of polynomials: for every $k$, the coefficient of $x^k$ is the same in all polynomials of degree at least $k$. 
A nondifferentiable continuous function like $|x|$ can be approximated in the second sense but not the first. 
