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

then this is sufficient to prove 1-1.

Suppose is monotone. Now suppose by way of contradiction that is not 1-1 then there exists such that without loss of generality assume that then by the monotone property we have that

but this is a contradiction!

Onto is false consider any piecewise increasing function with jump discontinuities.