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Question Number 272 by 123456 last updated on 25/Jan/15
$$\mathrm{given}\:\mathrm{f}\left(\mathrm{x},{y}\right)={ax}+{bxy}+{cy}\:\mathrm{for}\:\left(\mathrm{x},{y}\right)\in\mathbb{R}^{\mathrm{2}} \\ $$$$\mathrm{proof}\:\mathrm{that}\:\mathrm{if}\:\mid{a}−{c}\mid\leqslant\mathrm{1}\:\mathrm{then}\:\mid{f}\left({x},{y}\right)−{f}\left({y},{x}\right)\mid\leqslant\mid{x}−{y}\mid \\ $$$$\mathrm{is}\:\mathrm{f}\left(\mathrm{x},{y}\right)\:\mathrm{bounced}? \\ $$
Answered by prakash jain last updated on 18/Dec/14
$${f}\left({x},{y}\right)−{f}\left({y},{x}\right)={ax}+{bxy}+{cy}−{ay}−{bxy}−{cx} \\ $$$$=\left({a}−{c}\right){x}−\left({a}−{c}\right){y}=\left({a}−{c}\right)\left({x}−{y}\right) \\ $$$$\mid{f}\left({x},{y}\right)−{f}\left({y},{x}\right)\mid=\mid\left({a}−{c}\right)\left({x}−{y}\right)\mid=\mid{a}−{c}\mid\mid{x}−{y}\mid \\ $$$$\mathrm{Since}\:\mid{a}−{c}\mid\leqslant\mathrm{1} \\ $$$$\mid{f}\left({x},{y}\right)−{f}\left({y},{x}\right)\mid\leqslant\mid{x}−{y}\mid \\ $$$${f}\left({x},{y}\right)\:\mathrm{is}\:\mathrm{not}\:\mathrm{bounded}.\: \\ $$