For define a step function on the interval by
Then, define
- Calculate .
- Find all values of such that .
- We calculuate:
- .
Note: There is an error in the book. The answers in the back of the book claim that , which is incorrect.
For define a step function on the interval by
Then, define
Note: There is an error in the book. The answers in the back of the book claim that , which is incorrect.
Draw the graph of on the interval .
The second to last line follows from this exercise (I.4.7, #6)
Draw the graph of for .
The final equality follows from here (I.4.7, #5)
Prove
Proof. From this exercise (1.11 #4, part b) we know
But, since is constant on the open subintervals of the partition
which contains every integer between and , we have on the open subintervals of (since there are no integers in the open subintervals). Hence, . Thus,
Find a step function such that
Let
Then,
as requested.
Compute the following integrals where is the greatest integer less than or equal to .
We define a characteristic function, , on a set of points on by
Let be a step functions taking the (constant) value on the th open subinterval of a partition of . Prove that for each , we have
Proof. First, we note that the open subintervals of some partition of are necessarily disjoint since . Hence, if then for exactly one .
So, we have
for all , and for any . Further, by definition of , we know if . So,
for each
Draw the graph of each function defined below.