The family $\{z^k\}$ on $[0,1]$ is not equicontinuous

Take \(f_k(x)=x^k\) on \([0,1]\) and test equicontinuity at the endpoint \(1\). Set \(\varepsilon=1/2\). Given any \(\delta>0\), choose \(x\in[0,1)\) with \(|1-x|<\delta\). Since \(x<1\), choose \(k\) large enough that \(x^k<1/2\). Then \[ |f_k(1)-f_k(x)|=1-x^k>{1\over2}=\varepsilon. \] Thus no single \(\delta\) works for the whole family at \(1\), so \(\{f_k\}\) is not equicontinuous.