  # 3. Recall Dirichlet's functionif E Qg(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.

Question help_outlineImage Transcriptionclose3. 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. fullscreen
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Step 1

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

Step 2 help_outlineImage Transcriptioncloseg(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. fullscreen

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