Approximation by polynomials with only real critical points

We strengthen the Weierstrass approximation theorem by proving that any real-valued continuous function on an interval I ⊂ R can be uniformly approximated by a real-valued polynomial whose only (possibly complex) critical points are contained in I . The proof uses a perturbed version of the Chebyshev polynomials and an application of the Brouwer fixed point theorem.