A.Let be the function defined on the reals by f(x)= x sin(1/x) if x is irrational;f(x)=0 if x is rational. (i)Prove that f is continuous at 0.(ii) Prove that f is discontinuous at 1. (iii) Is f...

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A.Let be the function defined on the reals by f(x)= x sin(1/x) if x is irrational;f(x)=0 if x is rational. (i)Prove that f is continuous at 0.(ii) Prove that f is discontinuous at 1. (iii) Is f continuous or not at 1/pi? Prove your answer.

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Answered Same DayDec 22, 2021

Answer To: A.Let be the function defined on the reals by f(x)= x sin(1/x) if x is irrational;f(x)=0 if x is...

Robert answered on Dec 22 2021

124 Votes

A.(i) To show that f is continuous at 0, we have to show that if {xn} is a sequence which converges to 0, then
{f(xn)} → f(0). So
Case 1: Let {xn} be a sequence of irrationals which converge to 0. Then we have f(0) = 0 since 0 ∈ Q. We
now have to show that f(xn) = xn · sin
(
1
xn
)
→ 0. But this is clear since xn → 0
Case 2: Now if...

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A.Let be the function defined on the reals by f(x)= x sin(1/x) if x is irrational;f(x)=0 if x is rational. (i)Prove that f is continuous at 0.(ii) Prove that f is discontinuous at 1. (iii) Is f...

Answer To: A.Let be the function defined on the reals by f(x)= x sin(1/x) if x is irrational;f(x)=0 if x is...

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