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

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?

Solutions

Expert Solution

Colby Messinger answered 1 year ago

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