Exercise 8, § 8

Prove now that there cannot exist a non-constant mapping $\psi$ from an interval $I \subset \mathbf{R}$ into the set $\mathbf{N}^\mathbf{N}$ of sequences of natural integers, which, for every $x \in I$, has the following property: for every integer $n$, there exists a neighbourhood $V$ of $x$ in $I$ such that, for every $y \in V$, the first $n$ terms of the sequence $\psi(y)$ are identical to the first $n$ terms of the sequence $\psi(x)$ (note that this would entail the continuity of $\psi$, when each of the factors $\mathbf{N}$ is provided with the discrete topology, and $\mathbf{N}^\mathbf{N}$ with the product of these topologies).