Question

The function \( f(x) \) is defined as follows:  \[ f(x) = \begin{cases} 2+x, & \text{if } x \geq 0 \\ 2-x, & \text{if } x \leq 0 \end{cases} \] Then function \( f(x) \) at \( x=0 \) is: 

• A. Continuous and differentiable.
• B. Continuous but not differentiable.
• C. Differentiable but not continuous.
• D. Neither continuous nor differentiable.

Hint:

A function is differentiable only if left-hand and right-hand derivatives are equal.

Answer 1

The Correct Option is B

We are given a piecewise function: \[ f(x) = \begin{cases} 2 + x, & \text{if } x \geq 0 \\ 2 - x, & \text{if } x \leq 0 \end{cases}\]

Step 1: Check continuity at \( x = 0 \)
Left-hand limit: \( \lim_{x \to 0^-} f(x) = 2 - 0 = 2 \)
Right-hand limit: \( \lim_{x \to 0^+} f(x) = 2 + 0 = 2 \)
Since both limits and \( f(0) = 2 \), function is continuous at \( x = 0 \). 

Step 2: Check differentiability at \( x = 0 \)
Left-hand derivative: \( \frac{d}{dx}(2 - x) = -1 \)
Right-hand derivative: \( \frac{d}{dx}(2 + x) = 1 \)
Since left-hand and right-hand derivatives are not equal, function is not differentiable at \( x = 0 \).

Correct answer: (B) Continuous but not differentiable.

Answer 2

Checking continuity: \[ \lim\limits_{x \to 0^-} f(x) = \lim\limits_{x \to 0^+} f(x) = 2 \] Since \( f(0) = 2 \), the function is continuous. Checking differentiability: \[ f'(x) = \begin{cases} 1, & x > 0 -1, & x < 0 \end{cases} \] Since left and right derivatives are not equal, \( f(x) \) is not differentiable at \( x = 0 \). 

Conclusion: The function is continuous but not differentiable.