Which one of the following functions is differentiable for all real values of x?

Options

A. `(x)/(|x|)`
B. `x|x|`
C. `(1)/(|x|)`
D. `(1)/(x)`

Explanation

To determine which function is differentiable for all real values of \( x \), we will analyze the given functions step by step.

### Step 1: Analyze the first function \( f(x) = \frac{x}{|x|} \)

1. **Definition**: The function \( f(x) = \frac{x}{|x|} \) is defined as:
- \( f(x) = 1 \) for \( x > 0 \)
- \( f(x) = -1 \) for \( x < 0 \)
- \( f(0) \) is undefined (since \( |x| = 0 \) at \( x = 0 \))

2. **Continuity Check**:
- The function is not defined at \( x = 0 \), hence it is not continuous at \( x = 0 \).

3. **Differentiability Check**:
- Since the function is not continuous at \( x = 0 \), it cannot be differentiable there.
- Therefore, \( f(x) = \frac{x}{|x|} \) is not differentiable for all real values of \( x \).

### Step 2: Analyze the second function \( f(x) = x |x| \)

1. **Definition**: The function \( f(x) = x |x| \) can be expressed as:
- \( f(x) = x^2 \) for \( x \geq 0 \)
- \( f(x) = -x^2 \) for \( x < 0 \)

2. **Continuity Check**:
- At \( x = 0 \):
- \( f(0) = 0 \)
- \( \lim_{x \to 0^-} f(x) = 0 \)
- \( \lim_{x \to 0^+} f(x) = 0 \)
- Since both one-sided limits equal \( f(0) \), the function is continuous at \( x = 0 \).

3. **Differentiability Check**:
- For \( x < 0 \): \( f'(x) = -2x \)
- For \( x > 0 \): \( f'(x) = 2x \)
- At \( x = 0 \):
- Right-hand derivative: \( f'(0^+) = 2(0) = 0 \)
- Left-hand derivative: \( f'(0^-) = -2(0) = 0 \)
- Since the left-hand and right-hand derivatives at \( x = 0 \) are equal, \( f(x) = x |x| \) is differentiable at \( x = 0 \).

### Conclusion

Since \( f(x) = x |x| \) is continuous and differentiable for all real values of \( x \), the correct answer is:

**The function \( f(x) = x |x| \) is differentiable for all real values of \( x \).**

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