The number of discontinuities of the greatest integer function f(x)=[x-1], `x in (-11/2 , 105)` is equal to

Answer

To find the number of discontinuities of the greatest integer function \( f(x) = [x - 1] \) over the interval \( x \in \left(-\frac{11}{2}, 105\right) \), we can follow these steps:

### Step 1: Identify the interval
The interval is given as \( x \in \left(-\frac{11}{2}, 105\right) \). We can convert \( -\frac{11}{2} \) into decimal form:
\[ -\frac{11}{2} = -5.5 \]
Thus, the interval can be rewritten as \( x \in (-5.5, 105) \).

### Step 2: Determine the integer values in the interval
Next, we need to find the integer values within the interval \( (-5.5, 105) \). The integers in this range start from \( -5 \) and go up to \( 104 \) (since \( 105 \) is not included).

The integers are:
\[ -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, \ldots, 104 \]

### Step 3: Count the integers
To count the integers from \( -5 \) to \( 104 \):
- The smallest integer is \( -5 \).
- The largest integer is \( 104 \).

To find the total number of integers, we can use the formula for counting integers in a range:
\[ \text{Total integers} = (\text{largest integer} - \text{smallest integer}) + 1 \]
Substituting the values:
\[ \text{Total integers} = (104 - (-5)) + 1 = (104 + 5) + 1 = 109 + 1 = 110 \]

### Step 4: Identify discontinuities of the greatest integer function
The greatest integer function, \( f(x) = [x - 1] \), is discontinuous at integer values of \( x - 1 \). Therefore, we need to find the integer values of \( x \) in the interval \( (-5.5, 105) \):
- The discontinuities occur at \( x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, \ldots, 104 \).

### Step 5: Exclude the continuity point
Since the function \( f(x) \) is continuous at \( x = 1 \) (as shown in the video transcript), we need to exclude this point from our count of discontinuities.

### Step 6: Final count of discontinuities
Thus, the total number of discontinuities is:
\[ \text{Total discontinuities} = 110 - 1 = 109 \]

### Final Answer
The number of discontinuities of the function \( f(x) = [x - 1] \) in the interval \( (-\frac{11}{2}, 105) \) is **109**.

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