Write the number of points of discontinuity of `f(x)=[x]` in `[3,7]`.

Answer

To find the number of points of discontinuity of the function \( f(x) = [x] \) (the greatest integer function) in the interval \([3, 7]\), we can follow these steps:

### Step 1: Understand the Greatest Integer Function
The greatest integer function, denoted as \( [x] \), returns the largest integer less than or equal to \( x \). For example:
- \( [3] = 3 \)
- \( [4.5] = 4 \)
- \( [5] = 5 \)

### Step 2: Identify the Interval
We are interested in the interval \([3, 7]\). This means we will evaluate the function from \( x = 3 \) to \( x = 7 \).

### Step 3: Determine the Values of \( f(x) \) in the Interval
In the interval \([3, 7]\), the function behaves as follows:
- For \( x \) in \([3, 4)\), \( f(x) = 3 \)
- For \( x \) in \([4, 5)\), \( f(x) = 4 \)
- For \( x \) in \([5, 6)\), \( f(x) = 5 \)
- For \( x \) in \([6, 7)\), \( f(x) = 6 \)
- At \( x = 7\), \( f(x) = 7 \)

### Step 4: Identify Points of Discontinuity
The function \( f(x) \) is discontinuous at integer points where the function jumps from one integer value to another. In the interval \([3, 7]\), the points of discontinuity occur at:
- \( x = 4 \) (jumps from 3 to 4)
- \( x = 5 \) (jumps from 4 to 5)
- \( x = 6 \) (jumps from 5 to 6)
- \( x = 7 \) (jumps from 6 to 7)

### Step 5: Count the Points of Discontinuity
From the analysis, we can see that there are four points of discontinuity:
- \( x = 4 \)
- \( x = 5 \)
- \( x = 6 \)
- \( x = 7 \)

### Conclusion
Thus, the number of points of discontinuity of \( f(x) = [x] \) in the interval \([3, 7]\) is **4**.

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