- For , prove
- For , with , define
Draw the graph of on the interval .

- Find all real such that

*Proof.*Let . Then is a partition of and is constant on the open subintervals of . Further, for . So,The second to last line follows from this exercise (I.4.7, #6)

- The graph is:
- By inspection, we have, .

For part 6, I don’t think 5/2 is an answer, I got {x=1, \frac {1\pm {\sqrt {97}}} {4}}

It is the answer. To see this, plot the graph of the function 2(x-1) together with the above graph and see where they intersect. You can find a more detailed answer here:

https://math.stackexchange.com/questions/1064997/exercise-on-graphing-integral-of-floor-function