Solve it stepby step and please with clear handwritten. Be ready to follow thecomment
IntegrabilityTopic
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 commonfactor), g(x)=1/n; if x doesn't belong to rationalnumber, g(x)=0
Use the sequentialcriterion for continuity to prove that g is discontinuous at everyrational number in[0,1]
Question.2 g iscontinuous at any irrational point in[0,1]. Explain why g isRiemann integrable on[0,1] based on the following fact that
Suppose h:[a,b]?R iscontinuous everywhere except at a countable number of pointsin[a,b]. Then h is Riemann integrable on[a,b]
Question.3
Letf:[0,1]?R bedefined by f(x)=0 if x=0; f(x)=1 if 0c is arational number in [0,1]. Compute(f?g)(c). Nowsuppose c is an irrational number in[0,1].Compute(f?g)(c). Can you recognize the function f?g:[0,1]?R?
