If f(x)=|2-|x-3|| is non differentiable in `X in S`. Then value of `underset(x in S) sum (f(f(x))` is

Answer

To solve the problem, we need to analyze the function \( f(x) = |2 - |x - 3|| \) and determine where it is non-differentiable. We will then compute the sum \( \sum_{x \in S} f(f(x)) \) where \( S \) is the set of points where \( f(x) \) is non-differentiable.

### Step 1: Identify where \( f(x) \) is non-differentiable

The function \( f(x) = |2 - |x - 3|| \) will be non-differentiable at points where the expression inside the absolute value changes sign.

1. The inner absolute value \( |x - 3| \) is non-differentiable at \( x = 3 \).
2. The outer absolute value \( |2 - |x - 3|| \) is non-differentiable when \( 2 - |x - 3| = 0 \), which occurs when \( |x - 3| = 2 \). This gives us two more points:
- \( x - 3 = 2 \) → \( x = 5 \)
- \( x - 3 = -2 \) → \( x = 1 \)

Thus, the points of non-differentiability are \( x = 1, 3, 5 \). Therefore, the set \( S = \{1, 3, 5\} \).

### Step 2: Calculate \( f(x) \) at points in \( S \)

Now we will evaluate \( f(x) \) at each point in \( S \):

1. **For \( x = 1 \)**:
\[ f(1) = |2 - |1 - 3|| = |2 - 2| = |0| = 0 \]

2. **For \( x = 3 \)**:
\[ f(3) = |2 - |3 - 3|| = |2 - 0| = |2| = 2 \]

3. **For \( x = 5 \)**:
\[ f(5) = |2 - |5 - 3|| = |2 - 2| = |0| = 0 \]

Thus, we have:
- \( f(1) = 0 \)
- \( f(3) = 2 \)
- \( f(5) = 0 \)

### Step 3: Calculate \( f(f(x)) \)

Next, we need to evaluate \( f(f(x)) \) for each \( x \in S \):

1. **For \( x = 1 \)**:
\[ f(f(1)) = f(0) = |2 - |0 - 3|| = |2 - 3| = | -1 | = 1 \]

2. **For \( x = 3 \)**:
\[ f(f(3)) = f(2) = |2 - |2 - 3|| = |2 - 1| = |1| = 1 \]

3. **For \( x = 5 \)**:
\[ f(f(5)) = f(0) = 1 \quad \text{(as calculated previously)} \]

### Step 4: Compute the summation

Now we can sum the values of \( f(f(x)) \) for \( x \in S \):
\[ \sum_{x \in S} f(f(x)) = f(f(1)) + f(f(3)) + f(f(5)) = 1 + 1 + 1 = 3 \]

### Final Answer

The value of \( \sum_{x \in S} f(f(x)) \) is \( \boxed{3} \).