Let `f(x)=[x]{x^(2)}+[x][x^(2)]+{x}[x^(2)]+{x}{x^(2)}, AA x in [0, 10]`
`[.] and {.}` the greatest integer and fractional part functions respectively). The number of points of discontinuity of `f(x)` is

Answer

To solve the problem, we need to analyze the function \( f(x) = [x]{x^2} + [x][x^2] + \{x\}[x^2] + \{x\}\{x^2\} \), where \([.]\) denotes the greatest integer function and \({.}\) denotes the fractional part function. We are tasked with finding the number of points of discontinuity of this function for \( x \in [0, 10] \).

### Step-by-step Solution:

1. **Understanding the Components of the Function**:
- The function consists of the greatest integer function \([x]\) and the fractional part function \(\{x\}\).
- The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\).
- The fractional part function \(\{x\} = x - [x]\) gives the non-integer part of \(x\).

2. **Identifying Points of Discontinuity**:
- The greatest integer function \([x]\) is discontinuous at integer values of \(x\) (i.e., at \(x = 0, 1, 2, \ldots, 10\)).
- The fractional part function \(\{x\}\) is continuous everywhere but is affected by the discontinuities of \([x]\).

3. **Analyzing the Function**:
- The function \(f(x)\) is a combination of products of \([x]\), \(\{x\}\), \([x^2]\), and \(\{x^2\}\).
- Since \([x]\) and \([x^2]\) are both discontinuous at integer points, we need to check the behavior of \(f(x)\) around these points.

4. **Finding the Points of Discontinuity**:
- The points of discontinuity of \(f(x)\) will occur at the integers from 0 to 10.
- Specifically, we need to check the points \(x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\).

5. **Counting the Points**:
- There are 11 integer points in the interval \([0, 10]\): \(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\).
- Each of these points is a point of discontinuity for \(f(x)\).

### Conclusion:
The number of points of discontinuity of the function \(f(x)\) in the interval \([0, 10]\) is **11**.