Let $$f: [-2, 2] \to \mathbb{R}$$ be defined by $$f(x) = \begin{cases} x[x], & -2 < x < 0 \\ (x - 1)[x], & 0 \leq x \leq 2 \end{cases}$$ where $$[x]$$ denotes the greatest integer function. If $$m$$ and $$n$$ respectively are the number of points in $$(-2, 2)$$ at which $$y = |f(x)|$$ is not continuous and not differentiable, then $$m + n$$ is equal to _______.

To solve for the number of points where $$y = |f(x)|$$ is not continuous ($$m$$) and not differentiable ($$n$$) on the interval $$(-2, 2)$$, we break down the function $$f(x)$$ based on the greatest integer function $$[x]$$.

1. Define $$f(x)$$ Piecewise

• For $$x \in (-2, -1)$$: $$[x] = -2 \implies f(x) = -2x$$
• For $$x \in [-1, 0)$$: $$[x] = -1 \implies f(x) = -x$$
• For $$x \in [0, 1)$$: $$[x] = 0 \implies f(x) = (x-1)(0) = 0$$
• For $$x \in [1, 2)$$: $$[x] = 1 \implies f(x) = (x-1)(1) = x-1$$
• At $$x = 2$$: $$[x] = 2 \implies f(x) = (2-1)(2) = 2$$

2. Analyze $$|f(x)|$$ for Continuity ($$m$$)

We check the points where the definition changes: $$x = -1, 0, 1$$.

• At $$x = -1$$: $$\lim_{x \to -1^-} |-2x| = 2$$ and $$\lim_{x \to -1^+} |-x| = 1$$. Discontinuous.
• At $$x = 0$$: $$\lim_{x \to 0^-} |-x| = 0$$ and $$\lim_{x \to 0^+} |0| = 0$$. Continuous.
• At $$x = 1$$: $$\lim_{x \to 1^-} |0| = 0$$ and $$\lim_{x \to 1^+} |x-1| = 0$$. Continuous.

So, $$m = 1$$ (at $$x = -1$$).

3. Analyze $$|f(x)|$$ for Differentiability ($$n$$)

A function is not differentiable if it is discontinuous or has a "sharp corner" (LHD $$\neq$$ RHD).

• At $$x = -1$$: Discontinuous, so not differentiable.
• At $$x = 0$$: * LHD: $$\frac{d}{dx}|-x| = \frac{d}{dx}(x) = 1$$ (since $$x$$ is negative approaching 0)
• RHD: $$\frac{d}{dx}|0| = 0$$.
• LHD $$\neq$$ RHD, so not differentiable.
• At $$x = 1$$:
• LHD: $$\frac{d}{dx}|0| = 0$$
• RHD: $$\frac{d}{dx}|x-1| = 1$$
• LHD $$\neq$$ RHD, so not differentiable.

So, $$n = 3$$ (at $$x = -1, 0, 1$$).

4. Final Calculation

$$m + n = 1 + 3 = \mathbf{4}$$