Which of the following statements is always true? ([.] represents the greatest integer function. a) If f(x) is discontinuous then `|f(x)|` is discontinuous b) If `f(x)` is discontinuous then f(|x|) is discontinuous c) f(x)=[g(x)] is discontinous when `g(x) is an integer d) none of these

Options

A. if f(x) is discontinuous then |f(x)| is discontinuous
B. if f(x) is discontinuous , then f(|x|) is discontinuous .
C. f(x) =[g(x) ] is discontinuous , when g(x) is an integer
D. none of these

Answer

To analyze the given statements, we will evaluate each option one by one to determine which statement is always true.

### Given Statements:
1. **Option a:** If \( f(x) \) is discontinuous, then \( |f(x)| \) is discontinuous.
2. **Option b:** If \( f(x) \) is discontinuous, then \( f(|x|) \) is discontinuous.
3. **Option c:** \( f(x) = \lfloor g(x) \rfloor \) is discontinuous when \( g(x) \) is an integer.
4. **Option d:** None of these.

### Step-by-Step Solution:

#### Step 1: Analyze Option a
- **Statement:** If \( f(x) \) is discontinuous, then \( |f(x)| \) is discontinuous.
- **Counterexample:** Consider \( f(x) = x \) for \( x \neq 0 \) and \( f(0) = 1 \). Here, \( f(x) \) is discontinuous at \( x = 0 \). However, \( |f(x)| = |x| \) is continuous everywhere.
- **Conclusion:** This statement is **not always true**.

#### Step 2: Analyze Option b
- **Statement:** If \( f(x) \) is discontinuous, then \( f(|x|) \) is discontinuous.
- **Counterexample:** Let \( f(x) = \begin{cases} 1 & \text{if } x < 0 \\ 0 & \text{if } x = 0 \\ 1 & \text{if } x > 0 \end{cases} \). Here, \( f(x) \) is discontinuous at \( x = 0 \), but \( f(|x|) \) is continuous for all \( x \).
- **Conclusion:** This statement is **not always true**.

#### Step 3: Analyze Option c
- **Statement:** \( f(x) = \lfloor g(x) \rfloor \) is discontinuous when \( g(x) \) is an integer.
- **Explanation:** The greatest integer function \( \lfloor g(x) \rfloor \) is discontinuous at points where \( g(x) \) is an integer because it jumps at those points. If \( g(x) \) takes integer values, then \( \lfloor g(x) \rfloor \) will indeed be discontinuous at those points.
- **Conclusion:** This statement is **always true**.

#### Step 4: Analyze Option d
- Since we have found that option c is always true, option d (none of these) is incorrect.

### Final Answer:
The correct answer is **Option c**: \( f(x) = \lfloor g(x) \rfloor \) is discontinuous when \( g(x) \) is an integer.

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