Menu Close

lets-f-R-R-and-g-Z-R-such-that-f-x-g-x-x-Z-given-a-Z-then-proof-or-give-a-counter-example-that-lim-x-a-f-x-g-a-




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

Leave a Reply

Your email address will not be published. Required fields are marked *