We will consider some concepts in analysis, topology, measure theory and dynamical systems which are defined in terms of inverses, and try to see why they are defined so.
The first example of the strange custom of defining concepts through inverses appears in the definition of continuity. Let us consider the definition of a continuous function from reals to reals. The intuitive definition should be that we must be able to graph the function without taking the pen off the paper.
A ⟹ B can be read as "A is true only if B is true as well." (it can also be read as " if A is true, then B is true as well"). Using this, we try to formalize our intuition of a continuous function: if x is close to y, then f(x) is close to f(y). So, x ≈ y ⟹ f(x) ≈ f(y). So far, so good. Small changes in x produce only small changes in f(x).
The issue now is that you can make arbitrarily small changes in x.
The modern definition is based on that of Bernard Bolzano. It can be explained as a game between two players, draughtsman and encyclopedia. Since we do not want to offend humans of any gender, but we are fine with offending automata, let us designate them by names D and E, and address them with pronouns it and them. E, despite being an automata, contains all the information about f.
The draughtsman has finished drawing the graph of f until the point x. The pen is now at f(x). D is filled with doubts on whether it will be able to proceed without lifting the pen from the paper. E tries to fill it with hope, through an infinite dialogue.
D: I don't want to lift my pen off the paper when drawing the graph
further.
E: Well, what exactly do you mean by "taking the pen off the
paper"?
D: OK. For example, I don't want the graph to jump by more than 1
in the immediate vicinity of x
E: Hmm. From my knowledge of f, I know that if you don't move more
than δ1 from x, f stays within f(x) ± 1.
D: Nice. I'll continue, then.
(hesitates.)
D: But wait. I don't even want to jump by 1. I actually need f to
jump by less than 1/2.
E: Fine, if you don't move more than δ2, a quantity lesser
than δ1, f will not move vertically by more than f(x)
± 1/2.
D: I don't want to move even by 1/4.
...
If E can continue this process of reassurance indefinitely to meet stronger and stronger demands of D, then f can be extended from x without lifting the pen. This is what we mean when we say that f is continuous at x.
A function f mapping reals to reals is continuous at a point
x if
(∀ ε > 0) (∃ δ > 0) |x-y| < δ
⟹ |f(x) - f(y)| < ε.
A continuous function is one
which is continuous at every point.
We can generalize this by saying that f is continuous at x if every open set containing f(x) is contained within some f(B), where B is an open set containing x. (The dialogue between D and E will proceed along a similar line as given before.)
A function from a topological space (X, T1 ) to a topological space (Y, T2 ) is continuous if for every open set B in T2, f-1 (B) ∈ T1.
What if you consider the class of functions f: R → R such that the image of an open set is an open set? Will this be the class of continuous functions, or be a proper subset, or be incomparable with the class of continuous functions?