Answer
Proof given below.
Work Step by Step
Select any two different values, $x_{1}$ and $x_{2}$ from the domain of f.
If $x_{1}\neq x_{2}$, one must be greater than the other.
Without loss of generality, say that we chose $x_{1}$ to be the smaller number.
So, we have $x_{1}\lt x_{2}$.
If the function is increasing, then $\quad f(x_{1})\lt f(x_{2}).$
If the function is decreasing, then $\quad f(x_{1})\gt f(x_{2}).$
In either case, the inequality is strict, meaning that $f(x_{1})\neq f(x_{2})$
Therefore,
$ x_{1}\neq x_{2}\Rightarrow f(x_{1})\neq f(x_{2})$
QED.
