# A question about the definition of a strictly increasing function

So a definition of a strictly increasing function is $$~x_1 < x_2 \implies f(x_1) < f(x_2)$$.

Can this be extended to be a two-way implication, namely, $$~x_1 < x_2 \iff f(x_1) < f(x_2)$$?

Thanks!

Yes, it is a two-way implication. Suppose that $$f(x_1). Could we have $$x_1=x_2$$? No, because then $$f(x_1)=f(x_2)$$. Could we have $$x_1>x_2$$? No, because$$x_1>x_2\iff x_2So, we must have $$x_1.
Say $$f$$ is strictly increasing.
To show $$f(x_1), assume to the contrary $$f(x_1) and $$x_2\ge x_1$$.
But $$x_2>x_1$$ implies $$f(x_2)>f(x_1)$$, a contradiction,
and $$x_2=x_1$$ implies $$f(x_2)=f(x_1)$$, also a contradiction.