Number of points where $f (x) = ∣ ∣ x s g n (1 - x^{2}) ∣ ∣$ is non-differentiable is

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Solution

To find: Number of points where $f (x) = | x . s g n (1 - x^{2}) |$ is non- differentiable
For $x \in (- 1, 1) \to (1 - x^{2}) > 0 \to s g n (1 - x^{2}) = 1$
For $x = (- 1, 1) \to (1 - x^{2}) = 0 ⟹ s g n (1 - x^{2}) = 0$
For $x \in (- \infty, - 1) \cup (1, \infty) \to (1 - x^{2}) < 0 ⟹ s g n (1 - x^{2}) = - 1$
$⟹ f (x) = ⎧ ⎨ ⎩ \begin{matrix} | x |; x \in (- 1, 1) 0; x \in (- 1, - 1) | - x | = | x |; x \in (- \infty, - 1) \cup (1, \infty) \end{matrix}$
Hence, $f (x) = | x |$ with discontinuity at $x = - 1, & 1$
$⟹$ Total points of non-differentiability $= 3$

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