For the sake of simplicity, I will only consider the definition of a strictly increasing function:
"The function $f(x)$ is strictly increasing on an interval $K$ if $$\forall x_1, x_2 \in K, x_1 < x_2 \implies f(x_1)<f(x_2)." (1)$$
But then I came across this statement in my textbook:
"The function $f(x)$ is strictly increasing on an interval $K$ if and only if $$\forall x_1, x_2 \in K, x_1 \neq x_2 \implies \frac{f(x_2)-f(x_1)}{x_2-x_1}>0." (2)$$ I've tried to prove $(2)$ using definition $(1)$ but I could only prove the "if". I think we need more than just definition $(1)$ to prove the "only if", which is weird. Right?
And based on that statement in my textbook, I can prove that:
"The function $f(x)$ is strictly increasing on an interval $K$ if and only if $$\forall x_1, x_2 \in K, x_1 < x_2 \iff f(x_1)<f(x_2)." (3)$$ Statement $(3)$ is obviously more detailed than $(1)$.
So to sum up, I have two questions: Do I really need more than definition $(1)$ to prove $(2)$? And why don't we make definition $(1)$ more detailed, like statement $(3)$?
Sorry it's kinda chaotic in here but I can't find a better way to organize my ideas. Any help is appreciated.
