The function `f:RtoR` given by f(x)=|x|

Options

A. continuous as well as differentiable at x=0
B. not continuous but differentiable at x=0
C. continuous but not differentiable at x=0
D. neither continuous nor differentiable at x=0

Answer

To solve the problem of determining the continuity and differentiability of the function \( f(x) = |x| \), we will analyze the function step by step.

### Step 1: Define the function
The function \( f(x) = |x| \) can be expressed in piecewise form:
\[ f(x) = \begin{cases} -x & \text{if } x < 0 \\ 0 & \text{if } x = 0 \\ x & \text{if } x > 0 \end{cases} \]

### Step 2: Check for continuity
To check if \( f(x) \) is continuous at \( x = 0 \), we need to verify the following:
1. \( f(0) \) is defined.
2. The limit of \( f(x) \) as \( x \) approaches 0 exists.
3. The limit equals \( f(0) \).

Calculating \( f(0) \):
\[ f(0) = |0| = 0 \]

Calculating the limit as \( x \) approaches 0:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (-x) = 0 \]
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x = 0 \]

Since both one-sided limits equal \( f(0) \):
\[ \lim_{x \to 0} f(x) = 0 \]
Thus, \( f(x) \) is continuous at \( x = 0 \).

### Step 3: Check for differentiability
To check if \( f(x) \) is differentiable at \( x = 0 \), we need to find the derivative from both sides.

Calculating the derivative for \( x < 0 \):
\[ f'(x) = -1 \quad \text{(for } x < 0\text{)} \]

Calculating the derivative for \( x > 0 \):
\[ f'(x) = 1 \quad \text{(for } x > 0\text{)} \]

Now, we check the derivative at \( x = 0 \):
\[ f'(0^-) = -1 \quad \text{and} \quad f'(0^+) = 1 \]

Since \( f'(0^-) \neq f'(0^+) \), the derivative does not exist at \( x = 0 \).

### Conclusion
The function \( f(x) = |x| \) is continuous everywhere but not differentiable at \( x = 0 \).