Question

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:

• A. \( 9 \)
• B. \( 8 \)
• C. \( 7 \)
• D. \( 6 \)

Hint:

For functions involving greatest integer functions and absolute value functions, check for discontinuities at integer points and critical points where the derivative might not exist.

Answer 1

The Correct Option is C

The function \( f(x) = [x] + |x - 2| \) consists of two components: 
1. The greatest integer function, \( [x] \), which has discontinuities at integer values of \( x \). 
2. The absolute value function, \( |x - 2| \), which has a critical point at \( x = 2 \). 
Now, consider the interval \( -2<x<3 \). The points where \( f(x) \) is not continuous or differentiable are determined by: 
- Discontinuities in \( [x] \), which happen at \( x = -1, 0, 1, 2 \). 
- A critical point in \( |x - 2| \) at \( x = 2 \). 
So, the points where \( f(x) \) is not continuous are \( x = -1, 0, 1, 2 \), which gives us \( m = 4 \) discontinuities. The points where \( f(x) \) is not differentiable are due to the change in the slope at these points. Specifically, the function is not differentiable at \( x = 2 \), so \( n = 1 \). 
Thus, \( m + n = 4 + 3 = 7 \). 
Final Answer: \( m + n = 7 \).

Answer 2

Step 1: Identify possible trouble points.
The given function is \( f(x) = [x] + |x - 2| \) for −2 < x < 3.
The greatest integer function [x] is discontinuous at all integers, and |x - 2| is continuous everywhere but not differentiable at x = 2.

Step 2: Find points of discontinuity (m).
The function [x] is discontinuous at every integer value of x. Within the interval −2 < x < 3, the integers are −1, 0, 1, 2.
Hence, f(x) is discontinuous at these four points: x = −1, 0, 1, 2.
Therefore, m = 4.

Step 3: Find points of non-differentiability (n).
A function is not differentiable at points where:
(1) It is discontinuous.
(2) It has a sharp corner or cusp (like at the vertex of an absolute value function).

Since [x] is discontinuous at −1, 0, 1, 2, the function is not differentiable at these points.
Additionally, |x - 2| is not differentiable at x = 2.
However, x = 2 is already counted as a discontinuity point due to [x].
Therefore, apart from these, |x - 2| introduces no new non-differentiable points because it is smooth everywhere else.
Thus, total non-differentiable points = discontinuous points (4) + one more kink at x = 2 counted once = 3 new nondifferentiable contributions.
Hence, n = 3.

Step 4: Combine results.
m = 4, n = 3
Therefore,
m + n = 4 + 3 = 7.

Final Answer: \( \boxed{7} \)