`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 discontinuous
D. None of the above

Answer

To determine the conditions under which the sum of two functions \( f + g \) is continuous, we can analyze the continuity of the individual functions \( f \) and \( g \).

### Step-by-Step Solution:

1. **Understanding Continuity**:
A function is continuous at a point if the limit of the function as it approaches that point equals the value of the function at that point. For \( f + g \) to be continuous, we need to analyze the continuity of \( f \) and \( g \).

2. **Theorem**:
If both \( f \) and \( g \) are continuous functions, then \( f + g \) is also continuous. This is a fundamental theorem in calculus.

3. **Example of Discontinuous Functions**:
Let's consider two specific functions:
- \( f(x) = \lfloor x \rfloor \) (the greatest integer function, which is discontinuous at integer values)
- \( g(x) = \{ x \} \) (the fractional part function, which is also discontinuous at integer values)

4. **Finding \( f + g \)**:
Now, we calculate \( f + g \):
\[ f(x) + g(x) = \lfloor x \rfloor + \{ x \} \]
By definition, \( \{ x \} = x - \lfloor x \rfloor \). Therefore:
\[ f(x) + g(x) = \lfloor x \rfloor + (x - \lfloor x \rfloor) = x \]

5. **Continuity of \( f + g \)**:
The function \( x \) is continuous for all \( x \). Thus, even though both \( f \) and \( g \) are discontinuous, their sum \( f + g \) is continuous.

6. **Conclusion**:
The condition under which \( f + g \) can be continuous is that both functions \( f \) and \( g \) can be discontinuous, and their sum can still yield a continuous function. Therefore, the statement holds true that \( f + g \) may be continuous if both \( f \) and \( g \) are discontinuous.

### Summary of Conditions:
- If both \( f \) and \( g \) are continuous, then \( f + g \) is continuous.
- If both \( f \) and \( g \) are discontinuous, \( f + g \) can still be continuous (as shown in the example).
- If one function is continuous and the other is discontinuous, we cannot conclude anything about the continuity of \( f + g \).