What is the minimum of structure one needs to have on a set in order to be able to speak of continuity?
If f is a function defined on a subset of R—or, more generally, of Euclidean n-space R n —we say that f is continuous at x0 if “f(x) approaches f(x 0 ) as x approaches x 0 .” With ε and δ, this statement can be made sufficiently precise for mathematical purposes.
For each ε > 0, there is δ > 0 such that |f(x) - f(x 0 )| < ε for all x such that |x - x 0 | < δ.
Crucial for the definition of continuity thus seems to be that we can measure the distance between two real numbers (or, rather, two vectors in Euclidean n-space).
If we want to speak of continuity of functions defined on more general sets, we should thus have a meaningful way to speak of the distance between two points of such a set: this, in a nutshell, is the idea behind a metric space.
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