Let \(M<\infty\) and \(\mathcal{F} \subseteq C[a,b]\). Assume that each \(f \in \mathcal{F}\) is differentiable on \((a,b)\) and satisfies \(|f(a)| \leq M\) and \(|f'(x)| \leq M\) for all \(x \in (a,b)\). Prove that \(\mathcal{F}\) is equicontinuous on \([a,b]\).
Solution
1. Every \(f \in \mathcal F\) is Lipschitz with constant \(M\): \(|f(x) - f(y)| \le M|x - y|\) for all \(x, y \in [a,b]\).
mean value theorem — for \(x \ne y\), \(f(x) - f(y) = f'(c)(x - y)\) for some \(c\) between \(x\) and \(y\), and \(|f'(c)| \le M\).
2. \(\mathcal F\) is equicontinuous on \([a,b]\).
2.1. Given \(\eps > 0\), set \(\delta = \eps/M\).
\(M < \infty\) by hypothesis.
2.2. For \(|x - y| < \delta\): \(|f(x) - f(y)| \le M|x - y| < \eps\) for every \(f \in \mathcal F\).
2.1.
2.3. Q.E.D.
2 is exactly the definition of equicontinuity (\(\delta\) independent of \(x, y\), and \(f\)).
3. (Remark) each \(f \in \mathcal F\) is also uniformly bounded: \(|f(x)| \le M + M(b - a)\).
\(|f(x)| \le |f(a)| + |f(x) - f(a)| \le M + M|x - a| \le M + M(b - a)\) by 1.