Continuity of measurement of observables in Classical Mechanics

Question

In these lecture notes on QM by Simon Rhea & Richie Dadhley based on the first lecture Frederic Schuller, the following is mentioned on page-2:

Recall that in classical mechanics an observable is a map $F : Γ → R$, where $Γ$ is the phase space of the system, typically given by the cotangent space $T^{*}Q$ of some configuration manifold $Q$. The map is taken to be at least continuous with respect to the standard topology on $R$ and an appropriate topology on $Γ$, and hence if $Γ$ is connected, we have $F (Γ) = I ⊆ R$.

What is the basis of wanting the Observable to be continous w.r.t standard topology on $R$ and approproriate topology on $\Gamma$?

Answer 1

If you have small perturbations caused by e.g. the environment of the system, you want that these perturbations in the state only give a correspondingly small perturbation of the values of the observable in question.

If this were not the case, then basically a measurement (of a discontinuous observable) could just be a "random" output of the corresponding measurement device and as such the term "measurement" would be useless, i.e. one could not speak of an observable having a certain value (in some error bound) for a given system...