Question Number 1124 by 123456 last updated on 17/Jun/15
![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?](https://www.tinkutara.com/question/Q1124.png)
$$\mathrm{lets}\:{f}:\left[\mathrm{0},\mathrm{1}\right]\rightarrow\mathbb{R}\:\mathrm{given}\:\mathrm{by} \\ $$$${f}\left({x}\right)=\begin{cases}{{x}}&{{x}\in\mathbb{Q}}\\{\mathrm{1}−{x}}&{{x}\notin\mathbb{Q}}\end{cases} \\ $$$$\mathrm{is}\:{f}\:\mathrm{continuos}\:\mathrm{at}\:{x}=\mathrm{1}/\mathrm{2}? \\ $$$$\mathrm{is}\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{fdx}\:\mathrm{riemann}\:\mathrm{integrable}? \\ $$$$\mathrm{is}\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{fdx}\:\mathrm{lebesgue}\:\mathrm{integable}? \\ $$
Commented by prakash jain last updated on 20/Jun/15
$$\mathrm{Continuity} \\ $$$${f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{LHL},\:{x}=\frac{\mathrm{1}}{\mathrm{2}}−\epsilon \\ $$$$\mid\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}+\epsilon−\frac{\mathrm{1}}{\mathrm{2}}\mid<\delta\Rightarrow\epsilon<\delta \\ $$$$\mid\frac{\mathrm{1}}{\mathrm{2}}−\epsilon−\frac{\mathrm{1}}{\mathrm{2}}\mid<\delta\Rightarrow\epsilon<\delta \\ $$$$\mathrm{RHL},{x}=\frac{\mathrm{1}}{\mathrm{2}}+\epsilon \\ $$$$\mid\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}−\epsilon−\frac{\mathrm{1}}{\mathrm{2}}\mid<\delta\Rightarrow\epsilon<\delta \\ $$$$\mid\frac{\mathrm{1}}{\mathrm{2}}+\epsilon−\frac{\mathrm{1}}{\mathrm{2}}\mid<\delta\Rightarrow\epsilon<\delta \\ $$$${f}\left({x}\right)\:\mathrm{is}\:\mathrm{continuous}\:\mathrm{at}\:{x}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$