COMEDK · Maths · 25. Continuity and Differentiability
\(\text { If } f(x)=\left\{\begin{array}{cc}
x & , \quad 0 \leq x \leq 1 \\
2 x-1 & , \quad x>1
\end{array}\right. \text { then }\)
- A \(f\) is not continuous but differentiable at \(x=1\)
- B \(f\) is differentiable at \(x=1\)
- C \(f\) is discontinuous at \(x=1\)
- D \(f\) is continuous but not differentiable at \(x=1\)
Answer & Solution
Correct Answer
(D) \(f\) is continuous but not differentiable at \(x=1\)
Step-by-step Solution
Detailed explanation
To check continuity at \(x=1\), we evaluate the left-hand limit and right-hand limit. Left-hand limit: \(\lim_{x \to 1^{-}} f(x) = \lim_{x \to 1^{-}} x = 1\). Right-hand limit: \(\lim_{x \to 1^{+}} f(x) = \lim_{x \to 1^{+}} (2x - 1) = 2(1) - 1 = 1\). Since \(f(1) = 1\) and the…
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