Can every continuous function on \(\bar \DD\) be uniformly approximated by polynomials in the variable \(z\)?
Hint: compare to Weierstrass for the real interval.
Solution
No: polynomials are holomorphic and the uniform limit of holomorphic functions is holomorphic. However, \(f(z) \da \bar{z}\) is continuous on \(\bar\DD\) but not holomorphic.