If `f(x)=x+{-x}+[x]`, where [x] and {x} denotes greatest integer function and fractional part function respectively, then find the number of points at which f(x) is both discontinuous and non-differentiable in `[-2,2]`.

Answer

To solve the problem, we need to analyze the function \( f(x) = x - \{x\} + [x] \), where \([x]\) is the greatest integer function and \(\{x\}\) is the fractional part function.

### Step-by-Step Solution:

1. **Understanding the Functions**:
- 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. **Rewriting the Function**:
- We can express \(f(x)\) as:
\[ f(x) = x - (x - [x]) + [x] = x - x + [x] + [x] = 2[x] + 1 \]
- Thus, we have:
\[ f(x) = 2[x] + 1 \]

3. **Identifying Points of Discontinuity**:
- The function \([x]\) is discontinuous at all integer values of \(x\).
- Therefore, \(f(x)\) will also be discontinuous at these points.

4. **Finding the Integer Points in the Interval \([-2, 2]\)**:
- The integer points in the interval \([-2, 2]\) are: \(-2, -1, 0, 1, 2\).
- This gives us a total of 5 integer points.

5. **Determining Non-Differentiability**:
- A function is non-differentiable at points where it is discontinuous.
- Since \(f(x)\) is discontinuous at the integer points, it is also non-differentiable at these points.

6. **Conclusion**:
- The number of points at which \(f(x)\) is both discontinuous and non-differentiable in the interval \([-2, 2]\) is 5.

### Final Answer:
The number of points at which \(f(x)\) is both discontinuous and non-differentiable in \([-2, 2]\) is **5**.