Question

The real function \( y = \sin^2(|x|) \) is

• A. continuous for all \(x\)
• B. differentiable for all \(x\)
• C. not continuous at \(x = 0\)
• D. not differentiable at \(x = 0\)

Hint:

Check the continuity and differentiability separately: A function can be continuous at a point but not differentiable there.

Answer 1

The Correct Option is A, B

Step 1: Check for continuity.
The function \(y = \sin^2(|x|)\) is continuous everywhere because the sine function and the square of a continuous function are both continuous. The absolute value function \(|x|\) is also continuous, and thus the composition of these functions remains continuous. Step 2: Check for differentiability.
Although the function is continuous for all \(x\), it is not differentiable at \(x = 0\). This is due to the fact that the derivative of \(\sin^2(|x|)\) at \(x = 0\) does not exist because the left-hand and right-hand derivatives are not equal. Step 3: Conclusion.
The correct answer is (A) continuous for all \(x\), as the function is continuous at every point, including \(x = 0\), but it is not differentiable at \(x = 0\).