Question Number 1093 by 123456 last updated on 12/Jun/15
$$\mathrm{can}\:\mathrm{a}\:\mathrm{function}\:{f}\:\mathrm{be}\:\mathrm{such}\:\mathrm{that}\:\mathrm{for}\:{a}\in\mathbb{R} \\ $$$${f}:\mathrm{X}/\left\{{a}\right\}\rightarrow\mathbb{R},\left({a}−\delta,{a}+\delta\right)\in\mathrm{X},\mathrm{X}\subset\mathbb{R},\delta\in\mathbb{R},\delta>\mathrm{0} \\ $$$$\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:{f}\left({x}\right)=\infty \\ $$$$\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\mid{f}'\left({x}\right)\mid<\infty \\ $$
Answered by prakash jain last updated on 12/Jun/15
$$\mathrm{Let}\:\mathrm{us}\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\mid{f}\:'\left({x}\right)\mid={L}<\infty \\ $$$$\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{f}\left({x}−{h}\right)−{f}\left({x}\right)}{{h}}={L} \\ $$$$\mathrm{Given}\:{f}\left({a}−{h}\right)<\infty \\ $$$$\mathrm{If}\:{L}\:\mathrm{is}\:\mathrm{finite}\:{f}\left({a}\right)\:\mathrm{is}\:\mathrm{finite}:\:\:\mathrm{contradiction}. \\ $$$$\mathrm{So}\:\mathrm{L}\:\mathrm{is}\:\mathrm{infinite}\:\mathrm{or}\:\mathrm{does}\:\mathrm{not}\:\mathrm{exist}. \\ $$