What is the motivation for why a function should be independent of coordinates?
In the case of a general manifold I kind of get why, since one (usually) defines a function $f$ as a map from the manifold to the reals, i.e. $f:M\rightarrow\mathbb{R}$, and so in this sense it is manifestly coordinate independent (since $f$ has been defined without introducing any particular coordinate system). However, is there a reason in general (both heuristically and technically) why a function should be coordinate independent? In physics, the standard argument seems to be that a function is a scalar and so has no directional dependence, thus it should be invariant under rotations of coordinate systems. However, I'm unsure how this extends in generality (for example, why should it be true for a coordinate translation. Is it simply that coordinates are an artefact of the observer and so the value of the scalar function should not depend on the coordinates chosen, much like a vector is coordinate independent and this requires that its individual components should transform under coordinate transformations)?
I'm fairly new to the concept of differential geometry so I apologise for such a basic question, but hopefully someone can help me out.
Hmm. I haven't read through the whole comment discussion but I'm finding it hard to understand the question. A function is a function. By itself it has nothing to do with coordinates. Its on the level of sets: you just have domain, codomain and function.
So choosing coordinates can't change a function; the coordinates are extraneous.
It's like defining an object A. Once it's defined its defined. And then saying "pick a number". Did the number somehow change the object!?