Greatest Integer Function - A collection of Calculus problems

1. CALCULUS
Greatest integer function
A collection of exercises
The function [ ∙ ] ∶ ℝ ⟶ ℝ defined by [𝑥] = greatest integer not larger than 𝑥 is usually called the greatest integer
function.
1. Specific properties
Let 𝑥, 𝑦 ∈ ℝ. Show that:
1A. [𝑥 + 𝑦] = [𝑥] + [𝑦] ∨ [𝑥 + 𝑦] = [𝑥] + [𝑦] + 1.
1B. [2𝑥] = [𝑥] + [𝑥 +
2
1
].1
(Hint: use the previous result).
2. Image
2A. Find the image of [ ∙ ].
3. Graph
3A. Draw the graph of the greatest integer function over the interval [−4, 4].
4. Injectivity and Surjectivity
4A. Verify that the greatest integer function is neither injective nor onto.
5. Limits
5A. Consider 𝑎 ∈ ℤ.
What is [𝑎+] − [𝑎−]? Discuss its meaning.
(Remark: [𝑎+] = lim
𝑥⟶𝑎+
[𝑥] and [𝑎−] = lim
𝑥⟶𝑎−
[𝑥]).
5B. Use the definition of limit to show that lim
𝑥⟶+∞
[𝑥] = +∞.
Conclude that lim
𝑥⟶−∞
[𝑥] = −∞ (make 𝑦 = −𝑥 in 1A.) and therefore [ ∙ ] is not bounded.
6. Continuity
6A. Study the continuity of [ ∙ ].
7. Derivative and Antiderivative
7A. For what values is the derivative of [ ∙ ] defined?
Write the expression of that derivative.
7B. Can we use the derivative to study the monotonicity of the function [ ∙ ] on ℝ? Justify.
7C. Is it possible to find an antiderivative for [ ∙ ]? Justify.
8. Integrability
8A. Justify that the greatest integer function is integrable on any interval [0, 𝑛], 𝑛 ∈ ℕ.
8B. Prove that  


n nn
t
0 2
)1(
.
1
More generally, [𝑛𝑥] =  









1
0
n
k n
k
xx (Hermite’s identity)

2. More problems…
1. Let 𝑛 ∈ ℕ and 𝑥 ∈ ℝ.
Show that:
 












n
x
n
x
2. Let 𝑛 ∈ ℕ.
Prove that:
[√𝑛 + 1] − [√ 𝑛] = {
1, if 𝑛 is a perfect square
0, otherwise
3. Determine whether the following improper integral converges or diverges.
∫ (−1)[𝑒 𝑥]
+∞
0
𝑑𝑥
(Remark: [ ∙ ] denotes the greatest integer function).
4. Show that [√𝑛2 + 2𝑛] = 𝑛, for 𝑛 ∈ ℕ.
5. Determine whether the following series converges or diverges.










1
2
)1(
n
n
n
6. Find all values of 𝑥 that solve the equation below:
2
1
1
2
2






 xx
7. The function { ∙ } ∶ ℝ ⟶ ℝ defined by {𝑥} = 𝑥 − [𝑥] is called the fractional part function.
What are the possible values of {𝑥} + {−𝑥}?
8. Evaluate the following integral:
∫ {𝑥}2[𝑥]
5
3
9. Show that:
  1
!
!
1


n n
en
(Hint: start by using Taylor’s formula and then apply Abel’s summation by parts to evaluate the series).
Miguel Fernandes