Question Number 1192 by 123456 last updated on 13/Jul/15
$$\mathrm{lets}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{and}\:{g}:\mathbb{Z}\rightarrow\mathbb{R}\:\mathrm{such}\:\mathrm{that} \\ $$$${f}\left({x}\right)={g}\left({x}\right),{x}\in\mathbb{Z} \\ $$$$\mathrm{given}\:{a}\in\mathbb{Z},\:\mathrm{then}\:\mathrm{proof}\:\mathrm{or}\:\mathrm{give}\:\mathrm{a}\:\mathrm{counter}\:\mathrm{example}\:\mathrm{that} \\ $$$$\underset{{x}\rightarrow{a}} {\mathrm{lim}}{f}\left({x}\right)={g}\left({a}\right) \\ $$
Commented by prakash jain last updated on 13/Jul/15
$$\mathrm{Limit}\:\mathrm{may}\:\mathrm{not}\:\mathrm{even}\:\mathrm{exist}\:\mathrm{for}\:{f}\left({x}\right). \\ $$
Answered by prakash jain last updated on 14/Jul/15
$${g}\left({x}\left\{={x}\right.\right. \\ $$$${f}\left({x}\right)=\begin{cases}{{x}}&{{x}\in\mathbb{Z}}\\{\mathrm{0}}&{{x}\notin\mathbb{Z}}\end{cases} \\ $$