2. I did not really understand the attachment. Here is the idea of what you can to show. Let $\displaystyle I$ be an interval, say $\displaystyle [a,b]$ for simplicity sake (it can in fact be any interval, open, even unbounded but all the proofs are the same like here). And also it is given that $\displaystyle f$ is one-to-one and continous on $\displaystyle [a,b]$ (meaning it takes on a value exactly one time). We want to show it is strictly increasing (or decreasing). Pick any two points $\displaystyle x,y\in [a,b]$ with $\displaystyle x<y$. Say (without lose of generality) that $\displaystyle f(x) < f(y)$. Now argue that if $\displaystyle c<d$ then $\displaystyle f(c)<f(d)$ for all $\displaystyle c,d\in [a,b]$ by contradiction. And the contradiction is that if this was not the case then by the intermediate value theorem it is possible to find two points haviing the same value which is an impossibility since we are told the function is one-to-one.