Real Analysis Problem

Show that there is a countable set F of functions of form x--->a_0 + a_1cosx + a_2cos(2x) + ... +a_ncos(nx) such that every continuous real-valued function on [0,pi] is the uniform limit of a sequence of functions (f_n) in F.

Real Analysis - Applications of Mean Value Theorem

Show that if the roots of the polynomial p are all real, then the roots of p' are all real. If, in addition, the roots of p are all simple, then the roots of p' are all simple.

Discuss the differentiability of each of the following functions at all real numbers and find its derivative at those real numbers at which it is differentiable.

Discuss the differentiability of each of the following functions at all real numbers and find its derivative at those real numbers at which it is differentiable. See attached file for full problem description.

Convergence or Divergence of Integrals

The problems in the file submitted are from an undergraduate course in real Analysis. If you are able to work the problems, please detail any theorems or lemmas used in your solutions. The book we are using is titled "The Elements of Real Analysis" by Robert G. Bartle. We are working on derivatives and integrals, but have not started infinite series. In the problem attached below, there are 3 integrals. Please note that there are two questions to be answered. 1)To discuss the convergence or divergence of each integral and 2)To discuss whether or not the integrals are absolutely convergent. The problem also asks to give reasons to each answer.

Integration Proof

The problem in the file submitted is from an undergraduate course in Real Analysis. If you are able to work the problems, please detail any theorems or lemmas used in your solutions. The book we are using is titled "The Elements of Real Analysis" by Robert G. Bartle. We are working on derivatives and integrals, but have not started infinite series.