It is well-known that on a compact subset every smooth function can be approximated by polynomials (in the supremum norm). This is called the Stone-Weierstrass Approximation Theorem. However it is not as well-known that we cannot simply take uniformly spaced interpolation points.
The Runge function
It is a theorem that when we use Chebyshev nodes we can successfully approximate any absolutely continuous function.
As an exercise for Elm and the line-charts package, I decided to make a small demonstration.