# Explaining discontinuity in a greatest integer function

• Deathfish
In summary, a greatest integer function, also known as a floor function, rounds down a given number to the nearest integer. It is represented by the notation "⌊x⌋" or "floor(x)" and causes discontinuity when the input value is an integer. This creates a sudden jump or gap in the graph of the function. While the discontinuity cannot be removed, it can be accounted for using a piecewise function.
Deathfish
Homework Statement
Find the numbers, if any, where the function is discontinuous.

f(x) = [[x - 2]]

The attempt at a solution

function is discontinuous for all integer values of x.

I know that this is the obvious answer, however I am required to explain this in clear mathematical style, methodically explaining why this is so. Also I am not sure if this is the correct notation for presenting the answer. Anyone can help me out?

Hi Deathfish!

There are several (equivalent) ways to show continuity/discontinuity. Which ones have you seen and which do you prefer?

## 1. What is a greatest integer function?

A greatest integer function, also known as a floor function, is a mathematical function that rounds down a given number to the nearest integer.

## 2. How is a greatest integer function represented?

A greatest integer function is represented by the notation "⌊x⌋" or "floor(x)". The symbol "⌊⌋" is known as the floor brackets.

## 3. What causes discontinuity in a greatest integer function?

Discontinuity in a greatest integer function occurs when the input value is an integer. For example, the greatest integer function of 3 is 3, but the greatest integer function of 3.5 is 3. This sudden jump from 3 to 3.5 creates a discontinuity in the function.

## 4. How do you explain the concept of discontinuity in a greatest integer function?

Discontinuity in a greatest integer function can be thought of as a "break" or "gap" in the graph of the function. This occurs because the function cannot output any values between two integers, resulting in a gap in the graph.

## 5. Can discontinuity in a greatest integer function be removed?

No, discontinuity in a greatest integer function cannot be removed as it is a fundamental property of the function. However, the gap in the graph can be "filled in" by using a piecewise function that accounts for the discontinuity at the integer values.

Replies
3
Views
844
Replies
2
Views
2K
Replies
15
Views
6K
Replies
10
Views
2K
Replies
9
Views
2K
Replies
4
Views
3K
Replies
2
Views
2K
Replies
3
Views
2K
Replies
16
Views
3K
Replies
1
Views
1K