A function is defined as follows:
`f(x)={{:(-(x)/(sqrt(x^(2))),xne0),(0,x=0):}`
Which one of the following is correct in respect of the above function?

Options

A. A) f(x) is continuous at x = 0 but not differentiable at x = 0
B. B) f(x) is continuous as well as differentiable at x = 0
C. C) f(x) is discontinuous at x = 0
D. D) None of the above

Answer

To analyze the function \( f(x) \) defined as:

\[ f(x) = \begin{cases} -\frac{x}{\sqrt{x^2}} & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \]

we need to determine the continuity and differentiability of the function at the point \( x = 0 \).

### Step 1: Evaluate the function for \( x \neq 0 \)

For \( x \neq 0 \), we can simplify \( f(x) \):

\[ f(x) = -\frac{x}{\sqrt{x^2}} \]

Since \( \sqrt{x^2} = |x| \), we can rewrite the function as:

\[ f(x) = -\frac{x}{|x|} \]

### Step 2: Determine the value of \( f(x) \) for different cases

1. **Case 1: \( x > 0 \)**

Here, \( |x| = x \), so:

\[ f(x) = -\frac{x}{x} = -1 \]

2. **Case 2: \( x < 0 \)**

Here, \( |x| = -x \), so:

\[ f(x) = -\frac{x}{-x} = 1 \]

### Step 3: Find the left-hand limit as \( x \) approaches 0

To find the left-hand limit \( \lim_{x \to 0^-} f(x) \):

\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} 1 = 1 \]

### Step 4: Find the right-hand limit as \( x \) approaches 0

To find the right-hand limit \( \lim_{x \to 0^+} f(x) \):

\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} -1 = -1 \]

### Step 5: Check the continuity at \( x = 0 \)

For \( f(x) \) to be continuous at \( x = 0 \), the following must hold:

\[ \lim_{x \to 0} f(x) = f(0) \]

We have:

- \( \lim_{x \to 0^-} f(x) = 1 \)
- \( \lim_{x \to 0^+} f(x) = -1 \)
- \( f(0) = 0 \)

Since the left-hand limit (1) does not equal the right-hand limit (-1), and neither equals \( f(0) \) (0), we conclude that:

\[ \text{The function is not continuous at } x = 0. \]

### Step 6: Determine differentiability at \( x = 0 \)

Since the function is not continuous at \( x = 0 \), it cannot be differentiable at that point.

### Conclusion

The correct statement regarding the function \( f(x) \) is that it is **not continuous at \( x = 0 \)**, and therefore **not differentiable at \( x = 0 \)**.