To solve the problem, we need to analyze the equation \( x + |y| = 2y \) and evaluate the three statements given.
### Step 1: Analyze the equation
The equation can be split into two cases based on the value of \( y \):
1. **Case 1**: \( y \geq 0 \) (where \( |y| = y \))
\[
x + y = 2y
\]
Rearranging gives:
\[
x = 2y - y \implies x = y
\]
Therefore, when \( y \geq 0 \), we have \( y = x \).
2. **Case 2**: \( y < 0 \) (where \( |y| = -y \))
\[
x - y = 2y
\]
Rearranging gives:
\[
x = 2y + y \implies x = 3y \implies y = \frac{x}{3}
\]
Therefore, when \( y < 0 \), we have \( y = \frac{x}{3} \).
### Step 2: Determine the values of \( y \) based on \( x \)
From the analysis:
- For \( x \geq 0 \), \( y = x \).
- For \( x < 0 \), \( y = \frac{x}{3} \).
### Step 3: Evaluate the statements
1. **Statement 1**: "y as a function of x is not defined for all real x."
- **Analysis**: The function is defined for all real \( x \) since we have explicit equations for both cases. Thus, this statement is **not correct**.
2. **Statement 2**: "y as a function of x is not continuous at \( x = 0 \)."
- **Analysis**: At \( x = 0 \), both cases yield \( y = 0 \). Therefore, the function is continuous at \( x = 0 \). This statement is also **not correct**.
3. **Statement 3**: "y as a function of x is differentiable for all x."
- **Analysis**: To check differentiability, we need to find the derivatives:
- For \( x > 0 \), \( y = x \) gives \( \frac{dy}{dx} = 1 \).
- For \( x < 0 \), \( y = \frac{x}{3} \) gives \( \frac{dy}{dx} = \frac{1}{3} \).
- At \( x = 0 \), the left-hand derivative is \( \frac{1}{3} \) and the right-hand derivative is \( 1 \). Since these two derivatives are not equal, \( y \) is not differentiable at \( x = 0 \). Thus, this statement is **not correct**.
### Conclusion
All three statements are not correct. Therefore, the correct answer is that statements 1, 2, and 3 are all incorrect.
