Question

Solve it step by step and please with clear handwritten. Be ready to follow the comment

Integrability Topic

Question1. Let g : [0, 1] —> R be defined by if x=0, g(x)=1; if x=m/n (m and n are positive integer with no common factor), g(x)=1/n; if x doesn't belong to rational number, g(x)=0

Use the sequential criterion for continuity to prove that g is discontinuous at every rational number in[0,1]

Question.2 g is continuous at any irrational point in[0,1]. Explain why g is Riemann integrable on[0,1] based on the following fact that

Suppose h:[a,b]→R is continuous everywhere except at a countable number of points in[a,b]. Then h is Riemann integrable on[a,b]

Question.3

Letf:[0,1]→R be defined by f(x)=0 if x=0; f(x)=1 if 0<x<=1 we know that f is integrable on [0,1] Suppose c is a rational number in [0,1]. Compute(f◦g)(c). Now suppose c is an irrational number in[0,1]. Compute(f◦g)(c). Can you recognize the function f◦g:[0,1]→R?

Answer 1