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 \( f(x) = x + \{x\} - x + [x] \), where \( [x] \) is the greatest integer function (or floor function) and \( \{x\} \) is the fractional part function.

### Step 1: Rewrite the function
The fractional part function can be expressed as:
\[ \{x\} = x - [x] \]
Substituting this into the function gives:
\[ f(x) = x + (x - [x]) - x + [x] \]
This simplifies to:
\[ f(x) = x - [x] + [x] = x \]

### Step 2: Identify discontinuities
The function \( f(x) = x \) is continuous everywhere except at points where \( [x] \) changes value. The greatest integer function \( [x] \) is discontinuous at integer values. Therefore, \( f(x) \) will be discontinuous at integer points.

### Step 3: Find integer points in the interval \([-2, 2]\)
The integer points in the interval \([-2, 2]\) are:
- \(-2\)
- \(-1\)
- \(0\)
- \(1\)
- \(2\)

### Step 4: Check differentiability
The function \( f(x) = x \) is differentiable everywhere except at the points of discontinuity of \( [x] \). Since \( f(x) \) is continuous at all points except integers, it is also non-differentiable at these points.

### Conclusion
Thus, the points where \( f(x) \) is both discontinuous and non-differentiable in the interval \([-2, 2]\) are exactly the integer points:
- \(-2\)
- \(-1\)
- \(0\)
- \(1\)
- \(2\)

The total number of such points is **5**.

### Summary of Steps
1. Rewrite the function using the definitions of the greatest integer and fractional part functions.
2. Identify the points of discontinuity (integer points).
3. Count the integer points in the interval \([-2, 2]\).
4. Confirm that these points are also non-differentiable.