Let be a continuous and strictly increasing function. Consider the line . Prove that:

- .
- has a unique intersection point with the line above.

**Solution**

- As we can without loss of generality assume that , hence . Thus,
Similarly , we can prove that .

- The function is continuous and strictly decreasing. It follows from question (i.) that . Since it follows that there exists an such that which is unique due to the monotony of .