Exercise 1, § 2

Let $f$ be the real-valued function on $\mathbf{R}$ which is equal to 0 for $x \leq 0$, to $x$ for $0 \leq x \leq 1$, and to 1 for $x \geq 1$. Show that the sequence of real-valued functions $f_n(x) = f(nx - n^2)$ ($n \in \mathbf{N}$) is equicontinuous but not uniformly equicontinuous on $\mathbf{R}$, although it is formed of uniformly continuous functions.