Which of the following functions have finite number of points of discontinuity in R ( where, `[*]` represents greatest integer function ) ?

Options

A. tan x
B. x[x]
C. `(|x|)/(x)`
D. `sin [pi x]`

Answer

To determine which of the given functions have a finite number of points of discontinuity in \( \mathbb{R} \), we will analyze each function one by one.

### Given Functions:
1. \( \tan x \)
2. \( [x] \) (greatest integer function)
3. \( \frac{|x|}{x} \)
4. \( \sin([ \pi x ]) \)

### Step 1: Analyze \( \tan x \)
The function \( \tan x \) is known to be discontinuous at points where \( \cos x = 0 \). This occurs at:
\[ x = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
Since there are infinitely many such points, \( \tan x \) has an infinite number of points of discontinuity.

**Hint:** Identify the points where the denominator of the tangent function is zero.

### Step 2: Analyze \( [x] \)
The greatest integer function \( [x] \) is discontinuous at every integer point. Therefore, it has an infinite number of points of discontinuity (at each integer).

**Hint:** Recognize that the greatest integer function jumps at every integer value.

### Step 3: Analyze \( \frac{|x|}{x} \)
The function \( \frac{|x|}{x} \) can be defined as:
\[ \frac{|x|}{x} = \begin{cases} 1 & \text{if } x > 0 \\ \text{undefined} & \text{if } x = 0 \\ -1 & \text{if } x < 0 \end{cases} \]
This function is discontinuous only at \( x = 0 \) where it is undefined. Thus, it has exactly one point of discontinuity.

**Hint:** Check the definition of the function at critical points, particularly where it might be undefined.

### Step 4: Analyze \( \sin([ \pi x ]) \)
The function \( [\pi x] \) is discontinuous at points where \( \pi x \) is an integer, which happens at:
\[ x = \frac{k}{\pi} \quad (k \in \mathbb{Z}) \]
Since there are infinitely many integers \( k \), \( \sin([ \pi x ]) \) will also have an infinite number of points of discontinuity.

**Hint:** Consider the points where the argument of the sine function changes due to the greatest integer function.

### Conclusion
From the analysis:
- \( \tan x \): Infinite points of discontinuity.
- \( [x] \): Infinite points of discontinuity.
- \( \frac{|x|}{x} \): 1 point of discontinuity.
- \( \sin([ \pi x ]) \): Infinite points of discontinuity.

Thus, the only function with a finite number of points of discontinuity is:
\[ \frac{|x|}{x} \]

### Final Answer:
The function \( \frac{|x|}{x} \) has a finite number of points of discontinuity in \( \mathbb{R} \).