Prove a version of Theorem 3.1 when m is arbitrarily many times differentiable on [0, 1). Hint: Let g be an arbitrarily many times differentiable function on R such that it is zero outside [0, 1]. In...

Prove a version of Theorem 3.1 when m is arbitrarily many times differentiable on [0, 1).

Hint: Let g be an arbitrarily many times differentiable function on R such that it is zero outside [0, 1]. In the proof of Theorem 3.1, let aj denote the left end point of Aj . Put

                m(x) = ∞ j=1 g x − aj pj  .

Then m is arbitrarily many times differentiable on [0, 1). Follow the line of the proof of Theorem 3.1.