If f is an odd function and f(0) is defined, must f(0)=?

Answer

To solve the problem, we need to understand the properties of odd functions and how they relate to the value of the function at zero.

### Step-by-Step Solution:

1. **Definition of an Odd Function**:
An odd function \( f(x) \) satisfies the property that for every \( x \) in its domain, \( f(-x) = -f(x) \). This means that the graph of the function is symmetric about the origin.

2. **Evaluate at Zero**:
We want to find the value of \( f(0) \). According to the property of odd functions, we can substitute \( x = 0 \) into the odd function property:
\[ f(-0) = -f(0) \]

3. **Simplify**:
Since \( -0 \) is simply \( 0 \), we can rewrite the equation:
\[ f(0) = -f(0) \]

4. **Solve for \( f(0) \)**:
Now, we can add \( f(0) \) to both sides of the equation:
\[ f(0) + f(0) = 0 \]
This simplifies to:
\[ 2f(0) = 0 \]
Dividing both sides by 2 gives:
\[ f(0) = 0 \]

5. **Conclusion**:
Therefore, if \( f \) is an odd function and \( f(0) \) is defined, it must be that \( f(0) = 0 \).

### Final Answer:
\[ f(0) = 0 \]