Let $f: \mathbb{R} \rightarrow \mathbb{R}^{2}$ be defined by $f=(u, v)$, where $u$ and $v$ are defined by $u(0)=v(0)=0$ and, for $t \neq 0, u(t)=t^{2} \sin (1 / t)$ and $v(t)=t^{2} \cos (1 / t)$. Show that $f$ is differentiable on $\mathbb{R}$.

Show that for any real non-zero $a,\left\|f^{\prime}(a)-f^{\prime}(0)\right\|>1$, where we regard $f^{\prime}(a)$ as the vector $\left(u^{\prime}(a), v^{\prime}(a)\right)$ in $\mathbb{R}^{2}$.