Past exam of the metric spaces and complex analysis course of the University of Oxford / 2017 / 4 / b / iv / Solution

Take $g_n(z)=z^n/n$ and $g(z)=0$, for $n\ge 1$. These functions are entire andso convergence is uniform on the closed unit disc. But $g_n^{\prime }(z)=z^{n-1}$, givingfor every $n$. In particular $g_n^{\prime }(1)=1$ never approaches zero. On every strictly smaller disc the derivative does converge uniformly, consistent with part (iii). This is uniform holomorphic convergence without boundary derivative convergence.