Find the points of discontinuity on

[-3,5] for the function f(x)=[[x-1]], where the double bracket indicates the greatest integer function.

Take a look at the graph. There is a discontinuity at every integer value for x. y=n-1 for n <= x < n+1. Then the graph jumps to the next integer value for y.

To find the points of discontinuity for the function f(x) = [[x - 1]] on the interval [-3, 5], we need to identify the values of x where the function is not continuous.

The greatest integer function, represented by [[x]], gives us the largest integer less than or equal to x. In this case, [[x - 1]] will give us the largest integer less than or equal to x - 1.

Let's break down the interval [-3, 5] into smaller segments and check the behavior of the function in each segment.

1. We start with the interval [-3, -2]:
When x belongs to this interval, x - 1 will be less than or equal to -3 - 1 = -4.
The greatest integer function [[-4]] evaluates to -4.
Therefore, f(x) = -4 in this interval.

2. Next, we consider the interval [-2, -1]:
When x belongs to this interval, -2 - 1 will be less than or equal to -1.
The greatest integer function [[-3]] evaluates to -3.
Therefore, f(x) = -3 in this interval.

3. Moving to the interval [-1, 0]:
When x belongs to this interval, -1 - 1 will be less than or equal to 0.
The greatest integer function [[-2]] evaluates to -2.
Therefore, f(x) = -2 in this interval.

4. In the interval [0, 1]:
When x belongs to this interval, 0 - 1 will be less than or equal to -1.
The greatest integer function [[-1]] evaluates to -1.
Therefore, f(x) = -1 in this interval.

5. In the interval [1, 2]:
When x belongs to this interval, 1 - 1 will be less than or equal to 0.
The greatest integer function [[0]] evaluates to 0.
Therefore, f(x) = 0 in this interval.

6. In the interval [2, 3]:
When x belongs to this interval, 2 - 1 will be less than or equal to 1.
The greatest integer function [[1]] evaluates to 1.
Therefore, f(x) = 1 in this interval.

7. Finally, in the interval [3, 5]:
When x belongs to this interval, 3 - 1 will be less than or equal to 2.
The greatest integer function [[2]] evaluates to 2.
Therefore, f(x) = 2 in this interval.

Based on our analysis, we see that the function f(x) = [[x - 1]] is continuous within each interval of [-3, 5]. Therefore, there are no points of discontinuity within this interval.