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Prove that the greatest integer function is continuous at all the points except at integer points.

Aniket Singh , 5 Months ago
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Askiitians Tutor Team

Last Activity: 5 Months ago

To prove that the greatest integer function is continuous at all points except at integer points, we need to first understand the behavior of the greatest integer function, often denoted as floor(x) or **⌊x⌋.

Definition of Greatest Integer Function:
The greatest integer function ⌊x⌋ is defined as the greatest integer less than or equal to x. Mathematically,

⌊x⌋ = n, where n is an integer, and n ≤ x < n + 1.
Continuity of a Function:
A function is continuous at a point c if:

The function is defined at c.
The limit of the function as x approaches c exists.
The limit of the function as x approaches c is equal to the function value at c.
Proof:
Step 1: Continuity at Non-integer Points
Let c be any real number that is not an integer. This means c lies in the interval [n, n+1) for some integer n.

For any x in the interval (n, n+1), we have ⌊x⌋ = n.
As x → c (approaching c from both sides), the value of ⌊x⌋ will remain constant and equal to n (since x will stay within the interval where the floor of x is n).
Thus, lim (x → c) ⌊x⌋ = n and ⌊c⌋ = n (since c is within [n, n+1)).
Therefore, the limit of the function at c is equal to the value of the function at c, so ⌊x⌋ is continuous at all points that are not integers.
Step 2: Discontinuity at Integer Points
Now, let's examine what happens when c is an integer. Let c = m, where m is an integer.

Consider approaching m from the left. For x just slightly less than m, we have ⌊x⌋ = m - 1 (since x lies in the interval [m-1, m)).
Consider approaching m from the right. For x just slightly greater than m, we have ⌊x⌋ = m (since x lies in the interval [m, m+1)).
Thus, as x approaches m from the left, the limit of ⌊x⌋ is m - 1, and as x approaches m from the right, the limit of ⌊x⌋ is m.

Since the left-hand limit and the right-hand limit are not equal, lim (x → m) ⌊x⌋ does not exist, implying that ⌊x⌋ is not continuous at integer points.

Conclusion:
The greatest integer function ⌊x⌋ is continuous at all real numbers x except at integer points. At integer points, the function has a discontinuity because the left-hand and right-hand limits are not equal. Therefore, the greatest integer function is continuous at all points except at integer points.

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