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

Options

A. continuous but not differentiable
B. differentiable but not continuous
C. continuous and differentiable
D. discontinuous and not differentiable

Answer

To determine the properties of the function \( f(x) = |x| \) at \( x = 0 \), we will analyze its continuity and differentiability step by step.

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

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

Here, let \( c = 0 \).

- **Condition 1**: Evaluate \( f(0) \):
\[ f(0) = |0| = 0. \]
So, \( f(0) \) is defined.

- **Condition 2**: Evaluate \( \lim_{x \to 0} f(x) \):
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} |x| = 0. \]

- **Condition 3**: Check if the limit equals the function value:
\[ \lim_{x \to 0} f(x) = 0 = f(0). \]

Since all three conditions are satisfied, \( f(x) \) 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. The derivative \( f'(0) \) can be found using the definition of the derivative:
\[ f'(0) = \lim_{h \to 0} \frac{f(0 + h) - f(0)}{h} = \lim_{h \to 0} \frac{|h| - 0}{h} = \lim_{h \to 0} \frac{|h|}{h}. \]

Now, we need to evaluate this limit from both sides:

- **Limit from the right** (\( h \to 0^+ \)):
\[ \lim_{h \to 0^+} \frac{|h|}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1. \]

- **Limit from the left** (\( h \to 0^- \)):
\[ \lim_{h \to 0^-} \frac{|h|}{h} = \lim_{h \to 0^-} \frac{-h}{h} = -1. \]

Since the left-hand limit and the right-hand limit are not equal, the limit does not exist. Therefore, \( f(x) \) is not differentiable at \( x = 0 \).

### Conclusion

The function \( f(x) = |x| \) is continuous at \( x = 0 \) but not differentiable at that point. Thus, the correct answer is:

**Continuous and Non-Differentiable.**