f+g may be a continuous function, if:

Options

A. f is continuous and g is discontinuous
B. f is discontinuous and g is continuous
C. f and g both are continous
D. None of the above

Answer

To determine under what conditions the sum of two functions \( f + g \) is continuous, we can follow these steps:

### Step-by-Step Solution

1. **Understand the Definition of Continuity**: A function is continuous at a point \( c \) if:
- \( f(c) \) is defined.
- \( \lim_{x \to c} f(x) \) exists.
- \( \lim_{x \to c} f(x) = f(c) \).

2. **Consider the Functions \( f(x) \) and \( g(x) \)**: Let’s assume \( f(x) \) and \( g(x) \) are two functions. For the sum \( f(x) + g(x) \) to be continuous, we need to analyze the continuity of each function.

3. **Check Individual Continuity**:
- If \( f(x) \) is continuous at a point \( c \), then it satisfies the conditions mentioned above.
- Similarly, if \( g(x) \) is continuous at the same point \( c \), it also satisfies the conditions.

4. **Apply the Property of Continuous Functions**: A fundamental property of continuous functions states that the sum of two continuous functions is also continuous. Therefore:
- If both \( f(x) \) and \( g(x) \) are continuous at a point \( c \), then \( f(x) + g(x) \) is continuous at that point.

5. **Conclusion**: Thus, for \( f + g \) to be continuous, both \( f \) and \( g \) must be continuous functions. If either \( f \) or \( g \) is not continuous at a point, then \( f + g \) will not be continuous at that point.

### Final Statement
The sum \( f + g \) is continuous if both \( f \) and \( g \) are continuous functions.