Explaining discontinuity in a greatest integer function

Click For Summary
SUMMARY

The function f(x) = [[x - 2]] is discontinuous at all integer values of x. This conclusion arises from the properties of the greatest integer function, which exhibits jumps at each integer point. The discontinuity can be methodically explained using the definition of continuity, which requires that the limit of the function as x approaches a point equals the function's value at that point. The notation used in the discussion is appropriate for representing the greatest integer function.

PREREQUISITES
  • Understanding of the greatest integer function (floor function)
  • Knowledge of continuity and discontinuity in mathematical functions
  • Familiarity with limit definitions in calculus
  • Basic mathematical notation and terminology
NEXT STEPS
  • Study the properties of the greatest integer function and its discontinuities
  • Learn about the formal definition of continuity in calculus
  • Explore methods for proving discontinuity using limits
  • Review mathematical notation for functions and their properties
USEFUL FOR

Students studying calculus, mathematics educators, and anyone interested in understanding the properties of discontinuous functions.

Deathfish
Messages
80
Reaction score
0
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?
 
Physics news on Phys.org
Hi Deathfish! :smile:

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

Similar threads

A few questions to make sure I understand Fourier series
  • · Replies 3 ·
Replies
3
Views
2K
Understanding the solution to a calculus problem (removable discontinuities)
  • · Replies 2 ·
Replies
2
Views
2K
Example of a bounded, increasing, discontinuous function
  • · Replies 15 ·
Replies
15
Views
7K
Finding discontinuities in functions
  • · Replies 10 ·
Replies
10
Views
2K
Proving Discontinuity at Integers for f(x) = [x2]-[x]2
  • · Replies 9 ·
Replies
9
Views
2K
Is the given function continuous at x= 0? f(x) = {sin(1/x) if x≠0, 1
  • · Replies 4 ·
Replies
4
Views
4K
Can rational functions inside logarithms have removable discontinuities?
  • · Replies 2 ·
Replies
2
Views
2K
What about continuity and discontinuity of this function?
  • · Replies 3 ·
Replies
3
Views
2K
Removable Discontinuity of f(x)=(4-x)/(16-x^2)
  • · Replies 16 ·
Replies
16
Views
3K
Check my solution for discontinuous function
  • · Replies 1 ·
Replies
1
Views
2K