Continuity and Differentiability
Differentiability at Origin
JEE Main 2022
Grade 12
Question:
The function $f(x) = \begin{cases} \frac{\sin(x^2)}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases}$ at $x = 0$ is:
(1) continuous and differentiable
(2) continuous but not differentiable
(3) not continuous
(4) differentiable but not continuous
Step-by-Step Solution
Key Concept: Continuous: $\lim_{x \to 0} \sin(x^2)/x = \lim x \cdot \sin(x^2)/x^2 = 0 \cdot 1 = 0 = f(0)\checkmark$. Differentiable: $f'(0) = \lim_{h \to 0} \frac{\sin(h^2)/h}{h} = \lim_{h \to 0} \frac{\sin h^2}{h^2} = 1$. So differentiable with $f'(0) = 1$. Answer: (1).
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)