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given-f-x-y-ax-bxy-cy-for-x-y-R-2-proof-that-if-a-c-1-then-f-x-y-f-y-x-x-y-is-f-x-y-bounced-

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Question Number 272 by 123456 last updated on 25/Jan/15
given f(x,y)=ax+bxy+cy for (x,y)∈R^2 proof that if ∣a−c∣≤1 then ∣f(x,y)−f(y,x)∣≤∣x−y∣ is f(x,y) bounced?
givenf(x,y)=ax+bxy+cyfor(x,y)∈R2proofthatif∣a−c∣⩽1then∣f(x,y)−f(y,x)∣⩽∣x−y∣isf(x,y)bounced?
Answered by prakash jain last updated on 18/Dec/14
f(x,y)−f(y,x)=ax+bxy+cy−ay−bxy−cx =(a−c)x−(a−c)y=(a−c)(x−y) ∣f(x,y)−f(y,x)∣=∣(a−c)(x−y)∣=∣a−c∣∣x−y∣ Since ∣a−c∣≤1 ∣f(x,y)−f(y,x)∣≤∣x−y∣ f(x,y) is not bounded.
f(x,y)−f(y,x)=ax+bxy+cy−ay−bxy−cx=(a−c)x−(a−c)y=(a−c)(x−y)∣f(x,y)−f(y,x)∣=∣(a−c)(x−y)∣=∣a−c∣∣x−y∣Since∣a−c∣⩽1∣f(x,y)−f(y,x)∣⩽∣x−y∣f(x,y)isnotbounded.

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