Let $[x]$ denote the greatest integer function, and let $m$ and $n$ respectively be the numbers of the points where the function $f(x) = [x] + |x - 2|$, $-2 < x < 3$, is not continuous and not differentiable. Then $m + n$ is equal to:

Step 1: Analyze the components of the function

The given function $f(x)$ is a sum of two functions: the greatest integer function $[x]$ and the absolute value function $|x - 2|$. We need to analyze the continuity and differentiability of each component within the interval $(-2, 3)$.

Step 2: Identify points of discontinuity for $[x]$ and $|x-2|$

The greatest integer function $[x]$ is discontinuous at integer values. Within the interval $(-2, 3)$, these are $x = -1, 0, 1, 2$. The absolute value function $|x - 2|$ is continuous everywhere.

Step 3: Determine points of discontinuity for $f(x)$

Since $[x]$ is discontinuous at $x = -1, 0, 1, 2$ and $|x - 2|$ is continuous at these points, their sum $f(x)$ will also be discontinuous at these points. Thus, there are $m = 4$ points of discontinuity.

Step 4: Identify points of non-differentiability for $[x]$ and $|x-2|$

The greatest integer function $[x]$ is not differentiable at integer values, which are $x = -1, 0, 1, 2$ in the given interval. The absolute value function $|x - 2|$ is not differentiable at $x = 2$ because its derivative changes abruptly at this point.

Step 5: Determine points of non-differentiability for $f(x)$

A function is not differentiable at points where it is discontinuous. Therefore, $f(x)$ is not differentiable at $x = -1, 0, 1, 2$. Additionally, $f(x)$ is not differentiable at $x = 2$ because $|x-2|$ is not differentiable there. However, $x=2$ is already included in the points of discontinuity. So, the points of non-differentiability are $x = -1, 0, 1, 2$. Thus, there are $n = 4$ points of non-differentiability.

Step 6: Calculate $m + n$

We found $m = 4$ points where the function is not continuous and $n = 4$ points where the function is not differentiable. Adding these values gives the final result.

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