Continuity and Differentiability
Differentiability of Oscillating Functions
Premium Question
Grade 12
Question:
$f(x) = \begin{cases} x \sin(1/x), & x \neq 0 \\ 0, & x = 0 \end{cases}$. Which statement is TRUE?
(1) $f$ is continuous but not differentiable at $x = 0$
(2) $f$ is differentiable at $x = 0$
(3) $f$ is neither continuous nor differentiable at $x = 0$
(4) $f$ is differentiable everywhere
Step-by-Step Solution
Key Concept: Continuous at 0: $|f(x)| = |x||\sin(1/x)| \leq |x| \to 0\checkmark$. Differentiable: $f'(0) = \lim_{h \to 0} h \sin(1/h)/h = \lim \sin(1/h)$ — does NOT exist (oscillates between $-1$ and 1). Not differentiable at $x = 0$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)