In Munkres book Topology, Theorem 50.5, page 311, the author gives the following definition in a proof.
Let $f:X\to \Bbb R^n$ be a continuous function where $X$ is a compact metric space. Define $$\Delta(f)=\sup\{\operatorname{diam}f^{-1}(\{z\}):z\in f(X)\}.$$
And he claimed that the number $\Delta(f)$ measures how far $f$"deviates" from being injective, which is not really clear to me, as for example taking $X=[-1,1]$ and $f(x)=x^2$ then the diameter is $2$ at $z=1$, which is the maximum that functions deefined on $[-1,1]$ could possible be, while one could "oscillates" $f$ inside more to create "less injectivity". But that is fine for now as it hasn't been proved to be useful, but then I encountered a contradiction at one of his problems, quoted below.
Suppose that $X$ is a locally compact Hausdorff space with a countable basis, such that every compact subspace of $X$ has topological dimension at most $m$. Let $N=2m+1$. Given $X\to \Bbb R^N$ a continuous map and a compact subspace $C$ of $X$, define $$U_{\varepsilon}(C)=\{f:\Delta(f|_C)<\epsilon\}.$$ Show that $U_\varepsilon(C)$ is dense in $C(X,\Bbb R^N)$.
This is exercise $6$, question $(d)$ on page 315. But if I were to take $f(x)=x^2$ while $X=\Bbb R$, $C=[-1,1]$ I really doubt one could approximate $f$ with functions in $U_\varepsilon([-1,1])$. Intuitively functions in $U_\varepsilon([-1,1])$ are "partially monotonic" functions possibly with little oscillations at some places where the diameter of this oscillation domain would not exceed $\epsilon$. If one could choose a sequence of functions $f_n$ in $U_\varepsilon([-1,1])$ to approximate $f(x)=x^2$, then the diameter of the fiber $f_n^{-1}(\{1\})$ of this sequence of functions should really be getting closer to $2$, while $\varepsilon$ could be arbitary small.
Where am I wrong about this? Could anyone provides a thought of where I was wrong?