If f(x) is a real valued odd function , then which one of the following is incorrect ?

Options

A. `(f(x)-f(-x))/(2)` is an odd function.
B. `(f(x)+f(-x))/(2)` is an even function.
C. `[|f(x)|+2]` is an even function, `[*]` denotes the greatest integer function.
D. `(f(x)-f(-x))/(2)` is neither even nor odd.

Explanation

To determine which statement is incorrect given that \( f(x) \) is a real-valued odd function, we will analyze the properties of odd functions and evaluate the provided options.

### Step-by-Step Solution:

1. **Understanding Odd Functions**:
An odd function satisfies the property:
\[ f(-x) = -f(x) \]
for all \( x \).

2. **Evaluate Each Option**:
We will analyze each option to see if it adheres to the properties of odd functions.

**Option 1**: Let \( g(x) = f(x) - f(-x) \).
- Using the property of odd functions:
\[ g(x) = f(x) - (-f(x)) = f(x) + f(x) = 2f(x) \]
- Now, check if \( g(-x) = -g(x) \):
\[ g(-x) = f(-x) - f(x) = -f(x) - f(x) = -2f(x) = -g(x) \]
- Therefore, \( g(x) \) is also an odd function.

**Option 2**: Let \( g(x) = f(x) + f(-x) \).
- Using the property of odd functions:
\[ g(x) = f(x) + (-f(x)) = 0 \]
- The function \( g(x) = 0 \) is both even and odd.

**Option 3**: Let \( g(x) = \text{greatest integer}(\lvert f(x) \rvert) + 2 \).
- Check \( g(-x) \):
\[ g(-x) = \text{greatest integer}(\lvert f(-x) \rvert) + 2 = \text{greatest integer}(\lvert -f(x) \rvert) + 2 = \text{greatest integer}(\lvert f(x) \rvert) + 2 = g(x) \]
- Thus, \( g(x) \) is an even function.

**Option 4**: This option states that \( g(x) \) is neither even nor odd.
- Since we have established that options 1, 2, and 3 are correct, option 4 must be incorrect.

3. **Conclusion**:
The incorrect statement is **Option 4**.