The number of discontinuities of the greatest integer function `f(x)=[x], x in (-(7)/(2), 100)` is equal to

Options

A. 104
B. 100
C. 103
D. 101

Explanation

To find the number of discontinuities of the greatest integer function \( f(x) = [x] \) over the interval \( \left(-\frac{7}{2}, 100\right) \), we can follow these steps:

### Step 1: Identify the interval
The interval given is \( \left(-\frac{7}{2}, 100\right) \). We first convert \( -\frac{7}{2} \) to a decimal for easier understanding:
\[ -\frac{7}{2} = -3.5 \]
Thus, the interval can be rewritten as \( (-3.5, 100) \).

### Step 2: Determine the integers in the interval
Next, we need to identify the integers that fall within this interval. The integers greater than \( -3.5 \) start from \( -3 \) and go up to \( 99 \) (since \( 100 \) is not included).

The integers in the interval are:
\[ -3, -2, -1, 0, 1, 2, \ldots, 99 \]

### Step 3: Count the integers
Now, we will count the integers from \( -3 \) to \( 99 \):
- The first integer is \( -3 \).
- The last integer is \( 99 \).

To find the total number of integers from \( -3 \) to \( 99 \), we can use the formula for counting integers in a range:
\[ \text{Number of integers} = \text{Last integer} - \text{First integer} + 1 \]
Substituting the values:
\[ \text{Number of integers} = 99 - (-3) + 1 = 99 + 3 + 1 = 103 \]

### Step 4: Identify the points of discontinuity
The greatest integer function \( f(x) = [x] \) is discontinuous at every integer point. Therefore, the points of discontinuity in our interval are exactly the integers we counted.

### Conclusion
Thus, the number of discontinuities of the function \( f(x) = [x] \) in the interval \( \left(-\frac{7}{2}, 100\right) \) is:
\[ \boxed{103} \]