Analysis II | Part IB, 2002

Let f:RR2f: \mathbb{R} \rightarrow \mathbb{R}^{2} be defined by f=(u,v)f=(u, v), where uu and vv are defined by u(0)=v(0)=0u(0)=v(0)=0 and, for t0,u(t)=t2sin(1/t)t \neq 0, u(t)=t^{2} \sin (1 / t) and v(t)=t2cos(1/t)v(t)=t^{2} \cos (1 / t). Show that ff is differentiable on R\mathbb{R}.

Show that for any real non-zero a,f(a)f(0)>1a,\left\|f^{\prime}(a)-f^{\prime}(0)\right\|>1, where we regard f(a)f^{\prime}(a) as the vector (u(a),v(a))\left(u^{\prime}(a), v^{\prime}(a)\right) in R2\mathbb{R}^{2}.

Typos? Please submit corrections to this page on GitHub.