Are there functions that can’t be linearly/locally approximated? We always speak of the derivative as being the “best linear approximation”. And we also speak of linearizing. However, what does this really mean? For a given function $F$, what conditions on it make the claim “the derivative is the best linear approximation to $F$” true?
Are there functions that can’t be “locally linear” or locally approximated? If so, are these just mostly pathological, and we don’t have any interest in them (e.g. they don’t really show up in math)?
Are there important functions or mathematical objects that don’t really subject themselves well to the tools of analysis and approximation? (I understand this is a very broad and vague question.) I mean, there may be mathematical objects that we don’t know whether they are amenable to such efforts, but are there (important) objects where we are sure they definitely aren’t? Sort of like how abstract algebra/Galois theory showed the limitations of using radicals, giving rise to the notion of unsolvability?
 A: 
However, what does this really mean? For a given function F, what
  conditions on it make the claim “the derivative is the best linear
  approximation to F” true?

It is unconditionally true. It is well-explained in the link of the comment.

Are there functions that can’t be “locally linear” or locally
  approximated?

Yes, a lot of them.

If so, are these just mostly pathological, and we don’t have any
  interest in them (e.g. they don’t really show up in math)?

No, they show up a lot and lot and lots of times in math, physics, everywhere. 

Are there important functions or mathematical objects that don’t
  really subject themselves well to the tools of analysis and
  approximation?

Just consider a euclidean norm function : $f(x)=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}$. This is not differentiable at the origin. (i.e. cannot be linearly approximated) Without this, you will not even able to talk about what is distance between two given points. Some other nontrivial examples will be heaviside step function (this is not even continuous), and dirac-delta function (actually this is not even a function! If you are interested, look for the distribution theory).
