If a function `f:RtoR` satisfy the equation `f(x+y)=f(x)+f(y),AAx,y` and the function f(x) is continuous at x=0 , then

Options

A. f(x) is continuous for all positive real values of x
B. f(x) is continuous for all x
C. f(x) =0 for all x
D. None of the above

Answer

To solve the problem, we need to analyze the functional equation given by \( f(x+y) = f(x) + f(y) \) for all \( x, y \in \mathbb{R} \) and the condition that \( f(x) \) is continuous at \( x = 0 \).

### Step-by-Step Solution:

1. **Substituting Values**:
Let's start by substituting \( x = 0 \) and \( y = 0 \) into the functional equation:
\[ f(0 + 0) = f(0) + f(0) \]
This simplifies to:
\[ f(0) = 2f(0) \]

2. **Solving for \( f(0) \)**:
Rearranging the equation gives:
\[ f(0) - 2f(0) = 0 \implies -f(0) = 0 \implies f(0) = 0 \]

3. **Using Continuity**:
Since \( f(x) \) is continuous at \( x = 0 \), we can express the limit as \( x \) approaches \( 0 \):
\[ \lim_{x \to 0} f(x) = f(0) = 0 \]

4. **Considering \( f(x) \) for General \( x \)**:
Now, we can analyze the functional equation further. For any \( k \in \mathbb{R} \), consider the limit as \( x \) approaches \( k \):
\[ f(k + h) = f(k) + f(h) \]
where \( h \) approaches \( 0 \).

5. **Taking Limits**:
Taking the limit as \( h \to 0 \) gives:
\[ \lim_{h \to 0} f(k + h) = f(k) + \lim_{h \to 0} f(h) \]
Since we know \( \lim_{h \to 0} f(h) = 0 \), we have:
\[ f(k) = f(k) + 0 \implies f(k) = f(k) \]
This is trivially true and does not provide new information.

6. **Exploring the Form of \( f(x) \)**:
To find a specific form of \( f(x) \), we can assume \( f(x) \) is linear. Let's assume:
\[ f(x) = cx \]
for some constant \( c \).

7. **Verifying the Assumption**:
Substituting \( f(x) = cx \) into the original functional equation:
\[ f(x+y) = c(x+y) = cx + cy = f(x) + f(y) \]
This holds true for all \( x, y \).

8. **Conclusion**:
Since \( f(0) = 0 \) and \( f(x) \) is continuous at \( 0 \), we conclude that:
\[ f(x) = cx \]
where \( c \) is a constant. The function \( f(x) \) is continuous for all \( x \in \mathbb{R} \).

### Final Result:
Thus, the function \( f(x) \) can be expressed as:
\[ f(x) = cx \quad \text{for some constant } c \in \mathbb{R} \]