Consider function `f: R - {-1,1}-> R`. `f(x)=x/[1-|x|]` Then the incorrect statement is

Options

A. A. it is continuous at the origin
B. B. it is not derivable at the origin
C. C. the range of the function is R
D. D. f is continuous and derivable in its domain

Answer

To solve the problem, we need to analyze the function \( f(x) = \frac{x}{1 - |x|} \) and determine which of the given statements about the function is incorrect.

### Step 1: Define the function based on the absolute value
The function can be defined piecewise based on the value of \( x \):
- For \( x > 0 \): \( f(x) = \frac{x}{1 - x} \)
- For \( x = 0 \): \( f(0) = 0 \)
- For \( x < 0 \): \( f(x) = \frac{x}{1 + x} \)

### Step 2: Check continuity at \( x = 0 \)
To check if \( f(x) \) is continuous at \( x = 0 \), we need to evaluate the left-hand limit, right-hand limit, and the function value at \( x = 0 \).

- **Right-hand limit** as \( x \to 0^+ \):
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \frac{x}{1 - x} = \frac{0}{1} = 0 \]

- **Left-hand limit** as \( x \to 0^- \):
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} \frac{x}{1 + x} = \frac{0}{1} = 0 \]

- **Value at \( x = 0 \)**:
\[ f(0) = 0 \]

Since the left-hand limit, right-hand limit, and the function value at \( x = 0 \) are all equal, \( f(x) \) is continuous at \( x = 0 \).

### Step 3: Check differentiability at \( x = 0 \)
To check if \( f(x) \) is differentiable at \( x = 0 \), we need to compute the left-hand derivative and the right-hand derivative.

- **Left-hand derivative**:
\[ f'(0^-) = \lim_{h \to 0^-} \frac{f(0 + h) - f(0)}{h} = \lim_{h \to 0^-} \frac{\frac{h}{1 + h} - 0}{h} = \lim_{h \to 0^-} \frac{h}{h(1 + h)} = \lim_{h \to 0^-} \frac{1}{1 + h} = 1 \]

- **Right-hand derivative**:
\[ f'(0^+) = \lim_{h \to 0^+} \frac{f(0 + h) - f(0)}{h} = \lim_{h \to 0^+} \frac{\frac{h}{1 - h} - 0}{h} = \lim_{h \to 0^+} \frac{h}{h(1 - h)} = \lim_{h \to 0^+} \frac{1}{1 - h} = 1 \]

Since the left-hand derivative and right-hand derivative at \( x = 0 \) are equal, \( f(x) \) is differentiable at \( x = 0 \).

### Step 4: Analyze the range of the function
To determine the range of \( f(x) \):
- For \( x > 0 \), as \( x \) approaches 1, \( f(x) \) approaches infinity.
- For \( x < 0 \), as \( x \) approaches -1, \( f(x) \) approaches negative infinity.

Thus, the range of \( f(x) \) is \( \mathbb{R} \).

### Step 5: Evaluate the statements
Now we can evaluate the statements:
- **(A)** It is continuous at the origin: **True**
- **(B)** It is not derivable at the origin: **False** (it is derivable)
- **(C)** The range of the function is \( \mathbb{R} \): **True**
- **(D)** \( f \) is continuous and derivable in its domain: **True**

### Conclusion
The incorrect statement is (B): "It is not derivable at origin."

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