Properties of $f(a)-f(b)=a-b$ Let:
$f(a)-f(b)=a-b$ 
and $f(a)>f(b)$ and $f(a),f(b)>0$ and $a>b$
Based on the above facts is it sufficient enough to say that $f(a)=a$ and $f(b)=b$? 

Divide throughout by $a-b$ to get
$$\frac{f(a) - f(b)}{a-b} = 1$$
Notice how this resembles the gradient of a straight line graph. 
In other words, $f$ is a straight line with gradient $1$.
