[sheepscreek]

Is there a mathematical proof or basis to back what you’re saying?

[constantcrying]

That polynomials do not approximate continuous functions globally??

That is pretty obvious. Consider that every polynomial grows arbitrarily large, but not every continuous function does.

[gthompson512]

There is a simple corollary to Stone-Weierstrass that extends to infinite intervals, but requires the use of rational functions.