Let $f(x) = \begin{cases} x^n \sin\left(\frac{1}{x^2}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}$…

The key is recognizing that continuity and differentiability at $x = 0$ depend on how fast $x^n$ decays relative to the oscillations of $\sin(1/x^2)$. Since $|\sin(1/x^2)| \leq 1$ always, the $x^n$ factor acts as a "damping envelope"—for large enough $n$, it squeezes the oscillations to zero as $x \to 0$. Different values of $n$ will determine whether $f$ is merely continuous, differentiable, or has higher-order smoothness at the origin.

The key is recognizing that continuity and differentiability at $x = 0$ depend on how fast $x^n$ decays relative to the oscillations of $\sin(1/x^2)$. Since $|\sin(1/x^2)| \leq 1$ always, the $x^n$ factor acts as a "damping envelope"—for large enough $n$, it squeezes the oscillations to zero as $x \to 0$. Different values of $n$ will determine whether $f$ is merely continuous, differentiable, or has higher-order smoothness at the origin.