Explaining discontinuity in a greatest integer function

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The function f(x) = [[x - 2]] is discontinuous at all integer values of x. This is due to the nature of the greatest integer function, which jumps at each integer point. To explain this mathematically, one can analyze the limits from both sides of each integer and demonstrate that they do not equal the function value at that point. Various methods exist to illustrate continuity or discontinuity, and preferences for these methods can vary among individuals. Clear mathematical notation and structured explanations are essential for effectively communicating the reasoning behind the discontinuities.
Deathfish
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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?
 
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Hi Deathfish! :smile:

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

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