The set of all points, where the function `f(x) =x/(1+|x|)` is differentiable, is

Options

A. `(-infty, infty)`
B. `(-0,infty)`
C. `(-infty,0) cup (0,infty)`
D. none of these

Answer

To determine the set of all points where the function \( f(x) = \frac{x}{1 + |x|} \) is differentiable, we will analyze the function by breaking it down based on the absolute value.

### Step 1: Define the function without the absolute value
The function \( f(x) \) can be expressed in two cases based on the value of \( x \):

1. **Case 1:** When \( x \geq 0 \)
\[ f(x) = \frac{x}{1 + x} \]

2. **Case 2:** When \( x < 0 \)
\[ f(x) = \frac{x}{1 - x} \]

### Step 2: Check differentiability at \( x = 0 \)
To check if \( f(x) \) is differentiable at \( x = 0 \), we need to find the left-hand derivative and the right-hand derivative at this point.

#### Left-hand derivative at \( x = 0 \)
The left-hand derivative is given by:
\[ f'_{-}(0) = \lim_{h \to 0^-} \frac{f(0 + h) - f(0)}{h} \]
Substituting \( f(0 + h) \) for \( h < 0 \):
\[ f(0 + h) = \frac{h}{1 - h} \]
And since \( f(0) = 0 \):
\[ f'_{-}(0) = \lim_{h \to 0^-} \frac{\frac{h}{1 - h} - 0}{h} = \lim_{h \to 0^-} \frac{1}{1 - h} = 1 \]

#### Right-hand derivative at \( x = 0 \)
The right-hand derivative is given by:
\[ f'_{+}(0) = \lim_{h \to 0^+} \frac{f(0 + h) - f(0)}{h} \]
Substituting \( f(0 + h) \) for \( h > 0 \):
\[ f(0 + h) = \frac{h}{1 + h} \]
And since \( f(0) = 0 \):
\[ f'_{+}(0) = \lim_{h \to 0^+} \frac{\frac{h}{1 + h} - 0}{h} = \lim_{h \to 0^+} \frac{1}{1 + h} = 1 \]

### Step 3: Conclusion on differentiability
Since both the left-hand derivative and the right-hand derivative at \( x = 0 \) are equal to 1, \( f(x) \) is differentiable at \( x = 0 \).

### Step 4: Differentiability elsewhere
For \( x < 0 \) and \( x > 0 \), both expressions \( \frac{x}{1 - x} \) and \( \frac{x}{1 + x} \) are differentiable as they are both rational functions and do not have any points of discontinuity or non-differentiability.

### Final Answer
Thus, the function \( f(x) \) is differentiable for all \( x \in (-\infty, \infty) \).