lets-f-0-1-R-given-by-f-x-x-x-Q-1-x-x-Q-is-f-continuos-at-x-1-2-is-0-1-fdx-riemann-integrable-is-0-1-fdx-lebesgue-integable-

Question Number 1124 by 123456 last updated on 17/Jun/15

lets f : [0, 1] \to R given by

f (x) = {\begin{cases} x & x \in Q \\ 1 - x & x \notin Q \end{cases}

is f continuos at x = 1 / 2 ?

is \underset{0}{\int^{1}} f d x riemann integrable ?

is \underset{0}{\int^{1}} f d x lebesgue integable ?

Commented by prakash jain last updated on 20/Jun/15

Continuity

f (\frac{1}{2}) = \frac{1}{2}

LHL, x = \frac{1}{2} - ϵ

∣ 1 - \frac{1}{2} + ϵ - \frac{1}{2} ∣< δ \Rightarrow ϵ < δ

∣ \frac{1}{2} - ϵ - \frac{1}{2} ∣< δ \Rightarrow ϵ < δ

RHL, x = \frac{1}{2} + ϵ

∣ 1 - \frac{1}{2} - ϵ - \frac{1}{2} ∣< δ \Rightarrow ϵ < δ

∣ \frac{1}{2} + ϵ - \frac{1}{2} ∣< δ \Rightarrow ϵ < δ

f (x) is continuous at x = \frac{1}{2}