Consider the equation `x+|y|=2y`.
Which of the following statements are not correct?
yas a function of x is not defined for all real x.
yas a function of x is not continuous at x=0.
yas a function of x is differentiable for all x.
Select the correct answer using the code given below.

Options

A. 1 and 2 olny
B. 2 and 3 only
C. 1 and 3 only
D. 1, 2 and 3

Explanation

To solve the problem, we need to analyze the equation \( x + |y| = 2y \) and determine the validity of the given statements regarding \( y \) as a function of \( x \).

### Step 1: Rewrite the equation
We start with the equation:
\[ x + |y| = 2y \]
This can be rearranged to express \( |y| \) in terms of \( x \):
\[ |y| = 2y - x \]

### Step 2: Consider cases for \( y \)
Since \( |y| \) can take two forms depending on whether \( y \) is non-negative or negative, we will analyze both cases.

#### Case 1: \( y \geq 0 \)
If \( y \geq 0 \), then \( |y| = y \). Substituting this into the equation gives:
\[ y = 2y - x \]
Rearranging this, we find:
\[ x = y \quad \Rightarrow \quad y = x \quad \text{(for } y \geq 0\text{)} \]

#### Case 2: \( y < 0 \)
If \( y < 0 \), then \( |y| = -y \). Substituting this into the equation gives:
\[ -y = 2y - x \]
Rearranging this, we find:
\[ -x = 3y \quad \Rightarrow \quad y = -\frac{x}{3} \quad \text{(for } y < 0\text{)} \]

### Step 3: Define \( y \) as a function of \( x \)
From the two cases, we can define \( y \) as a piecewise function:
\[ y(x) = \begin{cases} x & \text{if } x \geq 0 \\ -\frac{x}{3} & \text{if } x < 0 \end{cases} \]

### Step 4: Analyze the statements
1. **Statement 1:** \( y \) as a function of \( x \) is not defined for all real \( x \).
- **Analysis:** This statement is incorrect because \( y \) is defined for all real \( x \).

2. **Statement 2:** \( y \) as a function of \( x \) is not continuous at \( x = 0 \).
- **Analysis:** To check continuity at \( x = 0 \):
- \( \lim_{x \to 0^-} y(x) = -\frac{0}{3} = 0 \)
- \( \lim_{x \to 0^+} y(x) = 0 \)
- \( y(0) = 0 \)
- Since both limits and the function value at \( x=0 \) are equal, \( y \) is continuous at \( x=0 \). This statement is incorrect.

3. **Statement 3:** \( y \) as a function of \( x \) is differentiable for all \( x \).
- **Analysis:** To check differentiability at \( x = 0 \):
- The left-hand derivative at \( x = 0 \) is \( \lim_{h \to 0^-} \frac{y(h) - y(0)}{h} = \lim_{h \to 0^-} \frac{-\frac{h}{3}}{h} = -\frac{1}{3} \)
- The right-hand derivative at \( x = 0 \) is \( \lim_{h \to 0^+} \frac{y(h) - y(0)}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1 \)
- Since the left-hand and right-hand derivatives are not equal, \( y \) is not differentiable at \( x = 0 \). This statement is also incorrect.

### Conclusion
All three statements are incorrect. Therefore, the correct answer is option D: all statements are not correct.