If `f(x) = [x] and g(x) = {x}=` fraction part of x, then for any two real numbers x and y.

Options

A. `f(x+ y) = f(x) + f(y)`
B. ` g (x + y ) = g (x) + g (y)`
C. `f (x + y ) = f (x) + f ( y + g(x))`
D. none of these

Explanation

To solve the problem, we need to analyze the functions \( f(x) = [x] \) (the greatest integer function) and \( g(x) = \{x\} \) (the fractional part function). We will check the validity of the given options based on these definitions.

### Step-by-Step Solution:

1. **Understanding the Functions**:
- The function \( f(x) = [x] \) returns the greatest integer less than or equal to \( x \).
- The function \( g(x) = \{x\} = x - [x] \) returns the fractional part of \( x \).

2. **Option 1: \( f(x+y) = f(x) + f(y) \)**:
- We need to check if \( [x+y] = [x] + [y] \) for any real numbers \( x \) and \( y \).
- This equality holds true only when both \( x \) and \( y \) are either integers or both have the same sign (both positive or both negative).
- Therefore, this option is **not true for all real numbers**.

3. **Option 2: \( g(x+y) = g(x) + g(y) \)**:
- We need to check if \( \{x+y\} = \{x\} + \{y\} \).
- The fractional part \( \{x+y\} \) can exceed 1 if both \( \{x\} \) and \( \{y\} \) are non-zero. For example, if \( x = 0.7 \) and \( y = 0.5 \), then \( \{x+y\} = \{1.2\} = 0.2 \), while \( \{x\} + \{y\} = 0.7 + 0.5 = 1.2 \).
- Hence, this option is also **not true for all real numbers**.

4. **Option 3: \( f(x+y) = f(x) + f(y) + g(x) \)**:
- We need to check if \( [x+y] = [x] + [y] + \{x\} \).
- We can express \( x \) and \( y \) as \( x = [x] + \{x\} \) and \( y = [y] + \{y\} \).
- Then, \( x + y = ([x] + \{x\}) + ([y] + \{y\}) = ([x] + [y]) + (\{x\} + \{y\}) \).
- The greatest integer of \( x+y \) can be expressed as \( [x+y] = [x] + [y] + [\{x\} + \{y\}] \).
- Since \( \{x\} + \{y\} \) can be greater than or equal to 1, we can write \( [\{x\} + \{y\}] = 1 \) when \( \{x\} + \{y\} \geq 1 \) and 0 otherwise.
- Thus, we can conclude that \( [x+y] = [x] + [y] + \{x\} \) holds true.
- Therefore, this option is **correct**.

### Conclusion:
The correct option is **Option 3**: \( f(x+y) = f(x) + f(y) + g(x) \).