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

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Advance Math

Solve it step by step and please with clear handwritten. Beready 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 integerwith no common factor), g(x)=1/n; if x doesn'tbelong to rational number, g(x)=0

Use the sequential criterion for continuity to prove that g isdiscontinuous 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 followingfact that

Suppose h:[a,b]?R is continuous everywhere except at a countablenumber 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 0c isa rational 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?

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