Clarification on the idea of absolute maxima

Question

$$f(x)=-|x|\:,\:\:x≠0$$ If f(x) did not have an point of discontinuity at x = 0, then it is obvious it would have an absolute maximum there. However, now that that point no longer exists, does it still have an absolute maximum? I’ve been told that there isn’t, but why not? Just because the point x = 0 doesn’t exist doesn’t mean the point an infinitesimal distance away from it doesn’t; in fact, two such points would exist for this function. Can’t those points be absolute maxima?

Answer 1

Yes, but how exactly do you "define" such a function?

You can't say something like $$\forall \varepsilon > 0, \ |x - 0| < \varepsilon $$ Since that would imply that $x = 0$.

However, if you pick some finite $\varepsilon$, then we can always pick something like $$\frac{\varepsilon}{2} < \epsilon$$ such that $$|x - \varepsilon/2| > |x - \varepsilon|$$