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Step-by-step Solution
Step 1: Understand the function
We are given the function
f(x) = \lvert 2 - \lvert x - 3 \rvert \rvert \quad \text{for } x \in \mathbb{R}.
We want to find all points where this function is not differentiable.
Step 2: Identify potential points of non-differentiability
A function involving absolute value(s) can fail to be differentiable at points where the argument of any absolute value becomes zero, or where the inside function is not differentiable. Here, there are two layers of absolute value:
Inner absolute value: \lvert x - 3 \rvert , which is not differentiable at x = 3 .
Outer absolute value: \lvert 2 - \lvert x - 3 \rvert \rvert , which is not differentiable if
2 - \lvert x - 3 \rvert = 0 \implies \lvert x - 3 \rvert = 2 \implies x - 3 = \pm 2 \implies x = 1 \text{ or } x = 5.
Thus, the set of points where f(x) is not differentiable is
S = \{\,1,\,3,\,5\}.
Step 3: Compute f(x) at each point in S
For x = 1 :
f(1) = \lvert 2 - \lvert 1 - 3 \rvert \rvert
= \lvert 2 - 2 \rvert
= 0.
For x = 3 :
f(3) = \lvert 2 - \lvert 3 - 3 \rvert \rvert
= \lvert 2 - 0 \rvert
= 2.
For x = 5 :
f(5) = \lvert 2 - \lvert 5 - 3 \rvert \rvert
= \lvert 2 - 2 \rvert
= 0.
Step 4: Compute f(f(x)) for each x \in S
We now evaluate f(f(x)) at each of these points:
For x = 1 :
f(1) = 0 \implies f(f(1)) = f(0).
Compute f(0) = \lvert 2 - \lvert 0 - 3 \rvert \rvert
= \lvert 2 - 3 \rvert
= 1.
Thus, f(f(1)) = 1.
For x = 3 :
f(3) = 2 \implies f(f(3)) = f(2).
Compute f(2) = \lvert 2 - \lvert 2 - 3 \rvert \rvert
= \lvert 2 - 1 \rvert
= 1.
Thus, f(f(3)) = 1.
For x = 5 :
f(5) = 0 \implies f(f(5)) = f(0) = 1.
Step 5: Summation of f(f(x)) over all x \in S
Therefore,
\sum_{x \in S} f(f(x))
= f(f(1)) + f(f(3)) + f(f(5))
= 1 + 1 + 1
= 3.
Final Answer
The required sum is 3 .
