Discontinuous derivative of a differentiable plane curve ($f(t)=t^2(\sin (1/t),\cos (1/t))$)

Set $f(0)=0$. The difference quotient has Euclidean norm $|t|$, so $f^{\prime }(0)=0$. For $t\ne 0$, differentiation gives $\Vert f^{\prime }(t){\Vert }^2=1+4t^2$. Thus this differentiable curve has a derivative discontinuous at zero, despite each scalar component satisfying the usual one-dimensional derivative properties.