- 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