Question

\( f(x) = \left\{ \begin{array}{ll} \sin \frac{1}{x} , & \text{for} \, x \neq 0 \\ 0, & \text{for} \, x = 0 \end{array} \right. \) is:

• A. continuous as well as differentiable
• B. differentiable but not continuous
• C. continuous but not differentiable
• D. neither continuous nor differentiable

Hint:

A function can be continuous at a point but not differentiable if it has oscillations or sharp corners at that point.

Answer 1

The Correct Option is C

Step 1: The function \( f(x) = \sin \frac{1}{x} \) is oscillating as \( x \to 0 \), which makes it non-differentiable at \( x = 0 \). However, it is continuous at \( x = 0 \).

Final Answer: \[ \boxed{\text{continuous but not differentiable}} \]