Which of the following is True?

Options

A. if `f(x)` is continuous at `x=c` and `g(x)` is discontinuous at `x=c` then `(f.g)(x)` must be discontinuous
B. If `f(x)` is continuous at `x=c` and `g(x)` is discontinuous at `x=c` then `(f.g)(x)` may be continuous.
C. If `f(x)` and `g(x)` are discontinuous at `x=c`, then the product function must be discontinuous.
D. If `f(x)` and `g(x)` are discontinuous at `x=c`, then the product function may be continuous.

Explanation

To determine which of the given statements is true, we will analyze each option step by step.

### Given Options:
1. If \( f(x) \) is continuous at \( x = c \) and \( g(x) \) is discontinuous at \( x = c \), then \( f(g(x)) \) must be discontinuous at \( x = c \).
2. If \( f(x) \) is continuous at \( x = c \) and \( g(x) \) is discontinuous at \( x = c \), then \( f(g(x)) \) may be continuous at \( x = c \).
3. If both \( f(x) \) and \( g(x) \) are discontinuous at \( x = c \), then \( f(g(x)) \) must be discontinuous at \( x = c \).
4. If both \( f(x) \) and \( g(x) \) are discontinuous at \( x = c \), then \( f(g(x)) \) may be continuous at \( x = c \).

### Step-by-Step Analysis:

**Step 1: Analyze Option 1**
- \( f(x) \) is continuous at \( x = c \).
- \( g(x) \) is discontinuous at \( x = c \).
- We need to check if \( f(g(x)) \) must be discontinuous at \( x = c \).
- Example: Let \( f(x) = x \) (continuous everywhere) and \( g(x) = \sin(\frac{1}{x}) \) for \( x \neq 0 \) and \( g(0) = 0 \) (discontinuous at \( x = 0 \)).
- Here, \( f(g(x)) = g(x) \) is still discontinuous at \( x = 0 \).
- However, if we take \( g(x) \) to be a function that approaches a limit (like \( g(x) = 0 \) at \( x = 0 \)), \( f(g(x)) \) can be continuous.
- **Conclusion**: This option is **not necessarily true**.

**Step 2: Analyze Option 2**
- \( f(x) \) is continuous at \( x = c \).
- \( g(x) \) is discontinuous at \( x = c \).
- We need to check if \( f(g(x)) \) may be continuous at \( x = c \).
- Using the same example as before, since \( f(x) = x \) is continuous, \( f(g(x)) \) can be continuous if \( g(x) \) approaches a specific limit.
- **Conclusion**: This option is **true**.

**Step 3: Analyze Option 3**
- Both \( f(x) \) and \( g(x) \) are discontinuous at \( x = c \).
- We need to check if \( f(g(x)) \) must be discontinuous at \( x = c \).
- Example: Let \( f(x) = 1 \) for \( x < 0 \) and \( 0 \) for \( x \geq 0 \) (discontinuous at \( x = 0 \)), and \( g(x) = 0 \) for \( x < 0 \) and \( 1 \) for \( x \geq 0 \) (discontinuous at \( x = 0 \)).
- Here, \( f(g(x)) = 1 \) for all \( x \) (continuous).
- **Conclusion**: This option is **not necessarily true**.

**Step 4: Analyze Option 4**
- Both \( f(x) \) and \( g(x) \) are discontinuous at \( x = c \).
- We need to check if \( f(g(x)) \) may be continuous at \( x = c \).
- Using the same example as in Option 3, we see that \( f(g(x)) \) can indeed be continuous.
- **Conclusion**: This option is **true**.

### Final Conclusion:
- The true options are **2 and 4**.
- Therefore, the correct statements are:
- **Option 2**: If \( f(x) \) is continuous at \( x = c \) and \( g(x) \) is discontinuous at \( x = c \), then \( f(g(x)) \) may be continuous.
- **Option 4**: If both \( f(x) \) and \( g(x) \) are discontinuous at \( x = c \), then \( f(g(x)) \) may be continuous.