Prove that the series

converges for all and let denote the value of this sum for each . Prove that is continuous for and prove that

*Proof.* First, the series converges for all real by the comparison test since

for all . Therefore, the convergence of implies the convergence of . Furthermore, this convergence is uniform by the Weierstrass -test with given by , and again converges. Thus, by Theorem 11.2 (page 425 of Apostol),

is continuous on the interval . Therefore, we may apply Theorem 11.4 (page 426 of Apostol):

since if and equals 2 if