one one and onto

• Mar 19th 2011, 08:35 AM
one 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 AM
TheEmptySet
Quote:

is monotonicity is sufficient condition for proving one one .do every function must satisfy this condition.
also what is best method to prove onto

If by monotone increasing you mean (the opposite for decreasing) the important part is the strict inequality.

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

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

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

Onto is false consider any piecewise increasing function with jump discontinuities.
• Mar 19th 2011, 09:09 AM
I think there is a slight bit of confusion going on here. It's true, for example, that for continuous $f:\mathbb{R}\to\mathbb{R}$ one has that strict monotonicity is equivalent to injectiveness.