Question
Asked Sep 28, 2019

Please just do 3(b). 

3. Recall Dirichlet's function
if E Q
g(x)
(a) Prove that g is not Riemann integrable on any interval [a, b] using the definition of integra-
bility
(b) Prove that g is not Riemann integrable on any interval [a, bj using Lebesgue's Theorem.
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3. Recall Dirichlet's function if E Q g(x) (a) Prove that g is not Riemann integrable on any interval [a, b] using the definition of integra- bility (b) Prove that g is not Riemann integrable on any interval [a, bj using Lebesgue's Theorem.

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Step 1

(b) To prove that the given function is not Riemann integrable using Lebesgue;s theorem

Step 2
g(x) 1xQ
0,x Q
g(x) is bounded (throughout R)
Claim g(x) discontinious at all x R.
Proof Let x Q
Let x} be a sequence of irrationals
with xxThen g(x,)01= g(x)
So,g(x) discontinious at x
Simiarly g(x)discontimious at xeQ.
help_outline

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g(x) 1xQ 0,x Q g(x) is bounded (throughout R) Claim g(x) discontinious at all x R. Proof Let x Q Let x} be a sequence of irrationals with xxThen g(x,)01= g(x) So,g(x) discontinious at x Simiarly g(x)discontimious at xeQ.

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