I am currently reading Chapter 15 of the book "Deep Learning Architectures, A Mathematical Approach" by Ovidiu Calin (see https://www.springer.com/de/book/9783030367206) and I am having trouble understanding the proposition about Translation Invariance of Pooling. More specifically, he introduces Max-Pooling of a continuous function $f: [a,b] \rightarrow \mathbb R$ with an equidistant partion $a = x_0 < ... < x_n = b$ of $[a,b]$ as $\mathcal{P}(f)(x) := \max_{x'\in [x_{i-1}, x_i]}f(x')$ for $x\in [x_{i-1}, x_i)$. Moreover he introduces translation as $T_a(x) := x- a$.
Proposition 15.2.1 then states:
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function. There is a partition of $\mathbb{R}$ such that $\mathcal{P}(T_a \circ f) = \mathcal{P}(f)$ for any small enough value of $a$.
In the proof he chooses a "finite partition $[x_i, x_{i+1}) 0\leq i \leq N-1$ such that the maxima $\xi_i$ of the restriction $f_{\vert [x_i, x_{i+1})}$ are inside the open intervals $(x_i, x_{i+1})$". How is that possible?
In my opinion, this does not make sense in two regards:
First, does he mean to use a partition of the form $(-\infty, x_0),[x_0, x_1),...,[x_N, \infty)$?
Secondly, the identity function, which obviously is continuous, does not fulfill $\mathcal P(T_a) = \mathcal P(id)$ for any nontrivial partition of $\mathbb R$ and maxima are never in the open interval $(x_{i-1}, x_{i})$.
Maybe someone knows how to put the assumptions right, such that the statements make sense?