1. Given function

We need to study

where $[x]$ is the greatest integer function.

We must find:

• $m$ = number of points where $f(x)$ is not continuous
• $n$ = number of points where $f(x)$ is not differentiable

Then compute $m+n$.

2. Where can problems occur?

The function is a sum of two parts:

• $[x]$, which is discontinuous at every integer
• $|x-2|$, which is continuous everywhere but not differentiable at $x=2$

So possible special points in the interval $(-2,3)$ are the integers inside it:

and also $x=2$ because of the modulus corner.

3. Continuity analysis

(i) Continuity of $[x]$

The greatest integer function has jump discontinuity at every integer. Thus in $(-2,3)$, discontinuities occur at

(ii) Continuity of $|x-2|$

This is continuous for all real $x$, including $x=2$.

Therefore, the sum $f(x)=[x]+|x-2|$ is discontinuous exactly at

So,

4. Differentiability analysis A function can be differentiable only where it is continuous. Hence all discontinuity points are automatically non-differentiable.

So $f(x)$ is not differentiable at least at

Now check whether there are any additional non-differentiable points.

(i) For $[x]$

Between consecutive integers, $[x]$ is constant, so differentiable there with derivative $0$.

(ii) For $|x-2|$

It is not differentiable at

but $x=2$ is already included above.

Thus there is no extra point besides the discontinuity points. Hence,

5. Compute $m+n$

6. Option check

The correct option is

which is Option C.

7. Comparison with stored answer

Stored correct answer: C

Our derived answer: C

So they agree.