Exercise 8, § 8

Show that there is no non-constant mapping $\psi$ of an interval $I \subset \mathbf{R}$ into the set $\mathbf{N}^\mathbf{N}$ of all sequences of natural numbers which has the following property for each $x \in I$: for each integer $n \geq 0$ there is a neighbourhood $V$ of $x$ in $I$ such that, for each $y \in V$, the first $n$ terms of the sequence $\psi(y)$ are the same as the first $n$ terms of the sequence $\psi(x)$ (remark that this would imply the continuity of $\psi$, if each factor $\mathbf{N}$ carries the discrete topology and $\mathbf{N}^\mathbf{N}$ the product topology).