Exercise 5.24. Define the Heaviside function or unit step function by rabalaodi l o if x 0. Let (ak) i be a sequence enumerating the rationals (i.e., every rational number appears exactly once as an ak), and define haga alvarl breed T(*) L H(x ax) 2k . k=1 (a) Prove the series f(x) converges absolutely for every real x, and the sum is between 0 and 1. OTTOMT 5.7. EXERCISESSAHD 109 (b) Show that for every real number x, the one-sided limit lim(f,x) is the sum of a taken over natural numbers k such that ak 0. Let (ak) i be a sequence enumerating the rationals (i.e., every rational number appears exactly once as an ak), and define haga alvarl breed T(*) L H(x ax) 2k . k=1 (a) Prove the series f(x) converges absolutely for every real x, and the sum is between 0 and 1. OTTOMT 5.7. EXERCISESSAHD 109 (b) Show that for every real number x, the one-sided limit lim(f,x) is the sum of a taken over natural numbers k such that ak