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# Consider the continuity of x(-1)^[1/x]

Define:

where denotes the greatest integer function, or floor function. Sketch the graph of for and . Evaluate

Is it possible to define in a way that makes continuous at 0.

First, we sketch the graph of on the requested intervals.

As , .
As , .

If we define then is continuous at 0.

# Consider the continuity of (-1)^[1/x]

Define:

where denotes the greatest integer function, or floor function. Sketch the graph of for and . Evaluate

Is it possible to define in a way that makes continuous at 0.

First, we sketch the graph of on the requested intervals.

As , alternates between and .
As , alternates between and .

There is no way to define to make continuous at 0 since will take both values and no matter how small we choose our . (So, if we were to try to define , then for there is no such that whenever , and similarly if we try to define .)

# Consider the limit and continuity of the floor function of (1/x)

Define:

where denotes the greatest integer function, or floor function. Sketch the graph of for and . Evaluate

Is it possible to define in a way that makes continuous at 0.

First, we sketch the graph of on the requested intervals.

As , takes on arbitrarily large positive values.
As , takes on arbitrarily large negative values.

There is no way to define to make continuous at 0.

# Establish some properties of the integral of the floor function

Define a function

where denotes the greatest integer less than or equal to , also called the floor function.

Then, define a function

1. Draw the graph of for and prove that is periodic with period 1.
2. For , prove

and prove that is also periodic with period 1.

3. Give a formula for in terms of the floor of .
4. Find a such that

5. For the constant from part (d), define

Prove that is periodic with period 1 and that if , we have,

1. The graph of is as follows:

Then, we prove that is periodic with period 1.
Proof. We compute to show (where I’ve replaced Apostol’s notation with since this less likely to cause confusion, also we use the solution to this exercise in the second line)

2. Proof. First, we establish the requested formula,

This was the requested formula. Next, to show is periodic with period 1, we show for any ,

Then, using the formula in the first part of (b) we have

Further, from part (a) we know is periodic with period 1, so , and we have,

by definition of . Thus, is periodic with period

3. To express in terms of of we compute as follows:

where since is an integer by definition, and has period 1, so for any integer . Continuing,

Here, we know the term in the integral is zero since for all since . Then,

This is the requested expression of in terms of .

4. Here we compute the integral, using the formula for we established in part (c), and solve for the requested constant ,

5. First we give the proof that is periodic with period 1.
Proof. We compute,

Next, we establish the requested formula. If , we have

as requested.

# Find some formulas for the integral of the step function [t]^2

1. For , prove

2. For , with , define

Draw the graph of on the interval .

3. Find all real such that

1. 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)

2. The graph is:

3. By inspection, we have, .

# Compute some integrals of step functions

1. Let , prove

2. Let , and define

Draw the graph of for .

1. Proof. Let be a partition of . Then, by the definition of the greatest integer function, is constant on the open subintervals of , so

The final equality follows from here (I.4.7, #5)

2. The graph is:

# Prove a formula for the sum of [(na)/b] where a,b are relatively prime integers

Let be relatively prime positive integers (i.e., they have no common factors other than 1). Then we have the formula

The sum is 0 when .

1. Prove this result by a geometric argument.
2. Prove this result by an analytic argument.

1. Proof. We know by the previous exercise (1.11, #6) that

Further, from this exercise (1.7, #4), we know

where number of interior lattice points, and number of boundary lattice points. We also know by the formula for the area of a right triangle that

Thus, we have,

Then, to calculate we note there are no boundary points on the hypotenuse of our right triangle (since and have no common factor). (This follows since if there were such a point then for some , but then we would have divides , contradicting that they have no common factor.) Thus, . So,

2. Proof. To derive the result analytically, first, by counting in the other direction we have,

Then,

# Formula for counting lattice points in the ordinate set of a function

For a nonnegative function defined on an interval define the set

(i.e., the region enclosed by the graph of the function and the -axis between the vertical lines at and ). Then prove

where is the greatest integer less than or equal to .

We can help ourselves by drawing a picture to get a good idea of what is going on, then turn that intuition into something more rigorous. The picture is as follows:

In the picture, we can see the number of lattice points in the ordinate set of (not including the -axis since the question stipulates contains the points ). At each integer between and , we count the number of lattice points beneath , the greatest integer less than or equal to . Then we need to turn this intuition from the picture into a proof:

Proof. Let with . We know such an exists since with . Then, the number of lattice points in with first element is the number of integers such that . But, by definition, this is (the greatest integer less than or equal to ). Summing over all integers we have,

# Deduce and prove a formula for [nx]

Use the previous exercise (1.11, #4 parts (d) and (e)) to deduce a formula for and prove this formula is correct.

Claim:

Proof. Let , then we have

Thus, there exists some with such that

Hence, . Thus,

Then, for each with we have

Then, for , we have

So,

# Prove some properties of the greatest integer function

For any we denote the greatest integer less than or equal to by . Prove the following properties of the function :

1. for any integer .
2. or .
3. .
4. .

1. Proof. Let for some integer . Then,

But, we defined . Thus, for any

2. Proof. If , then for some . Hence, and

On the other hand, if , then let . This gives us,

3. Proof. Let and , then we have

Thus,

4. Proof. By part (c) we have,

If , then let ,

Thus, .
On the other hand, if , then let , and

Thus,

5. Proof. By part (c) we have

And,

So, putting these together we have,

If , then, let , so

Thus, .
Next, if , then let , giving us,

Thus,

Finally, if , then let , and we have

So,