A question about the definition of a strictly increasing function

Yes, it is a two-way implication. Suppose that $f(x_1)<f(x_2)$. 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$$x_1>x_2\iff x_2<x_1\implies f(x_2)<f(x_1).$$So, we must have $x_1<x_2$.

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Definition

Real Analysis

Monotone Functions

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