Let \(\mcf\) be a family of differentiable functions \(f:[0,1]\to\CC\) such that \[ |f'(x)|\le M \] for every \(f\in\mcf\) and every \(x\in[0,1]\), for one constant \(M\) independent of \(f\). Show that \(\mcf\) is equicontinuous.
Hint: apply the MVT.
Solution
For every \(f\in\mcf\) and \(x,y\in[0,1]\), the mean-value estimate gives \[ |f(x)-f(y)|\le M|x-y|. \] Thus the whole family is uniformly Lipschitz with the same constant \(M\). Given \(\varepsilon>0\), if \(M>0\) take \(\delta=\varepsilon/M\); if \(M=0\) any \(\delta>0\) works. Then \(|x-y|<\delta\) implies \[ |f(x)-f(y)|<\varepsilon \] for every \(f\in\mcf\). Hence \(\mcf\) is equicontinuous.