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

{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.