Question Number 1462 by 123456 last updated on 07/Aug/15
$$\mathrm{can}\:\mathrm{a}\:\mathrm{function}\:{f}_{{n}} :\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{be}\:\mathrm{such}\:\mathrm{that} \\ $$$$\underset{{n}=\mathrm{0}} {\overset{+\infty} {\sum}}{f}_{{n}} \left({x}\right)\:\mathrm{converge}\:\mathrm{for}\:{x}\in\mathbb{Q}\:\mathrm{and}\:\mathrm{diverge} \\ $$$$\mathrm{for}\:{x}\in\mathbb{R}\backslash\mathbb{Q}? \\ $$
Commented by prakash jain last updated on 08/Aug/15
$$\mathrm{The}\:\mathrm{generic}\:\mathrm{nature}\:\mathrm{of}\:\mathrm{this}\:\mathrm{question}\:\mathrm{allows} \\ $$$$\mathrm{for}\:\mathrm{direct}\:\mathrm{examples}\:\mathrm{such}\:\mathrm{as} \\ $$$${f}_{{n}} \left({x}\right)=\begin{cases}{{g}_{{n}} \left({x}\right)}&{{x}\in\mathbb{Q}}\\{{h}_{{n}} \left({x}\right)}&{{x}\in\mathbb{R}\backslash\mathbb{Q}}\end{cases} \\ $$
Commented by 123456 last updated on 09/Aug/15
$$\mathrm{what}\:\mathrm{if}\:{f}_{{n}} \:\mathrm{have}\:\mathrm{to}\:\mathrm{be}\:\mathrm{continuous}? \\ $$