is monotonicity is sufficient condition for proving one one .do every function must satisfy this condition.

also what is best method to prove onto

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- Mar 19th 2011, 08:35 AMayushdadhwalone one and onto
is monotonicity is sufficient condition for proving one one .do every function must satisfy this condition.

also what is best method to prove onto - Mar 19th 2011, 08:49 AMTheEmptySet
If by monotone increasing you mean (the opposite for decreasing) the important part is the strict inequality.

$\displaystyle x_1 < x_2 \implies f(x_1) < f(x_2)$ then this is sufficient to prove 1-1.

Suppose $\displaystyle f$ is monotone. Now suppose by way of contradiction that $\displaystyle f$ is not 1-1 then there exists $\displaystyle x_1 \ne x_2$ such that $\displaystyle f(x_1)=f(x_2)$ without loss of generality assume that $\displaystyle x_1 < x_2$ then by the monotone property we have that

$\displaystyle f(x_2) = f(x_1) < f(x_2)=f(x_1)$ but this is a contradiction!

Onto is false consider any piecewise increasing function with jump discontinuities. - Mar 19th 2011, 09:09 AMayushdadhwal
sir good evening

sir suppose i choose a function with restricted domain{1,2,3,4) such that f(1)=2 f(2)=1 f(3)=6 f(4)=-3 (ie point graph will be formed)

now this function is one one but not neither decreasing nor increasing .little bit doubt sir please clear - Mar 20th 2011, 11:35 PMDrexel28