Question

If \(f(x) = \begin{cases} x, & 0 \le x \le 1 \\ 2x - 1, & 1<x \end{cases}\), then

• A. f is discontinuous at \(x = 1\)
• B. f is differentiable at \(x = 1\)
• C. f is continuous but not differentiable at \(x = 1\)
• D. None of the above

Hint:

Function continuous if left and right limits equal; differentiable if left and right derivatives equal.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Check continuity and differentiability at \(x = 1\).
Step 2: Detailed Explanation:
Left limit: \(f(1) = 1\)
Right limit: \(\lim_{x \to 1^+} (2x - 1) = 1\), so continuous
Left derivative: \(f'(1^-) = 1\)
Right derivative: \(f'(1^+) = 2\)
Derivatives not equal, so not differentiable.
Step 3: Final Answer:
Continuous but not differentiable at \(x = 1\).