The function f (x ) = |x | at x = 0 is

Options

A. continuous but non-differentiable
B. discontinuous and differentiable
C. discontinuous and non-differentiable
D. continuous and differentiable

Answer

To determine the properties of the function \( f(x) = |x| \) at \( x = 0 \), we need to check for continuity and differentiability.

### Step 1: Check for Continuity at \( x = 0 \)

A function is continuous at a point if the following three conditions are satisfied:
1. \( f(0) \) is defined.
2. The limit \( \lim_{x \to 0} f(x) \) exists.
3. \( \lim_{x \to 0} f(x) = f(0) \).

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

**Calculating the limit:**
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} |x| = 0 \]

**Comparing the limit and the function value:**
\[ \lim_{x \to 0} f(x) = 0 = f(0) \]

Since all three conditions are satisfied, \( f(x) = |x| \) is continuous at \( x = 0 \).

### Step 2: Check for Differentiability at \( x = 0 \)

A function is differentiable at a point if the derivative exists at that point. We can check the differentiability by finding the left-hand and right-hand derivatives at \( x = 0 \).

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

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

Since the left-hand derivative \( f'(0^-) = -1 \) and the right-hand derivative \( f'(0^+) = 1 \) are not equal, the derivative does not exist at \( x = 0 \).

### Conclusion

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

### Final Answer

The correct option is: Continuous but not differentiable.

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