For real x with `-10 le x le 10 ` define `f(x) = int_(-10)^(x) 2^([t]) dt` , where for a real number r we denote by [r] the largest integer less than or equal to r. The number of points of discontinutiy of f in the interval (10,10) is

Options

A. 0
B. 10
C. 18
D. 19

Answer

To solve the problem, we need to analyze the function defined as:

\[ f(x) = \int_{-10}^{x} 2^{[t]} dt \]

where \([t]\) denotes the greatest integer less than or equal to \(t\). We want to find the number of points of discontinuity of \(f\) in the interval \((-10, 10)\).

### Step-by-step Solution:

1. **Understanding the Function**:
The function \(f(x)\) is defined as an integral from \(-10\) to \(x\) of \(2^{[t]}\). The function \(2^{[t]}\) is piecewise constant because \([t]\) changes its value at every integer.

2. **Identifying Points of Discontinuity**:
The points of discontinuity for \(f(x)\) can occur where the integrand \(2^{[t]}\) changes its value, which happens at integer values of \(t\). Therefore, we need to check the integers in the interval \([-10, 10]\).

3. **Finding the Integers in the Interval**:
The integers in the interval \([-10, 10]\) are:
\(-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\).
This gives us a total of \(21\) integers.

4. **Evaluating Left and Right Limits**:
For each integer \(r\) in this interval, we need to evaluate the left-hand limit (LHL) and right-hand limit (RHL) of \(f(x)\) as \(x\) approaches \(r\).

- **Left-hand limit** as \(x \to r^{-}\):
\[ \text{LHL} = \int_{-10}^{r} 2^{[t]} dt \]
- **Right-hand limit** as \(x \to r^{+}\):
\[ \text{RHL} = \int_{-10}^{r} 2^{[t]} dt + \int_{r}^{r+h} 2^{[r]} dt \text{ (for small } h\text{)} \]
This will yield:
\[ \text{RHL} = \int_{-10}^{r} 2^{[t]} dt + h \cdot 2^{[r]} \]

5. **Checking Continuity**:
The function \(f(x)\) will be continuous at \(r\) if:
\[ \text{LHL} = \text{RHL} \]
This holds true because the integral from \(-10\) to \(r\) is the same in both cases, and the additional term \(h \cdot 2^{[r]}\) vanishes as \(h\) approaches \(0\).

6. **Conclusion**:
Since \(f(x)\) is continuous at each integer point \(r\) in the interval \([-10, 10]\), there are no points of discontinuity in the interval \((-10, 10)\).

Thus, the number of points of discontinuity of \(f\) in the interval \((-10, 10)\) is:

\[ \boxed{0} \]