# Continuous Functions

A function f:A->B has a natural representation as a Chu space M with height that of A and width that of B. For each point a in A, the a-th row of M is taken to be the f(a)-th row of B. (Hence the constraint on the width.) Each function f:A->B appearing as a single row of A-oB is just the M representing f, flattened to one row by reading M across and then down (row-major order).

We say that f:A->B is continuous when transposing M yields the representation of a function from B transposed to A transposed.

This latter function must also be continuous, because if we transpose all three spaces again we end up back where we were with M as the representation of a function from A to B, namely f. We denote this function from B_|_ to A_|_ by f_|_.

To say this with quantifiers, let X and Y be the set of column indices of A and B respectively, and let A and B do double duty as the respective sets of row indices (a common overloading in algebra). Write r(a,x) for the entry at row a and column y of A, and similarly s(b,y) for the (b,y)-th entry of B. We can then say that the Chu space A is the triple (A,r,X), B is (B,s,Y), and M is (A,m,Y) where m(a,y) = s(f(a),y) for all a in A and y in Y, and that f_|_:Y->X.

Then f:A->B is continuous when for every y in Y there exists x in X (namely f_|_(y)) such that for all a in A, r(a,x) = m(a,y). We can cut out the middle man M from this definition by replacing m(a,y) by s(f(a),y), these being equal. We end up with the condition s(f(a),y) = r(a,f_|_(y), which has now taken on the form of an adjointness condition.

For Chu spaces whose states are the open sets of a topological space, this definition of continuity turns out to coincide with the usual definition in terms of inverse images of open sets. The nice thing about continuity is that it is just as useful a notion when the usual requirements on open sets are dropped, namely that they be closed under arbitrary union and finite intersection.

The continuous functions from A to B turn out to be precisely those functions that preserve all properties of A, for a certain natural albeit abstract notion of property [Pr95]. This behavior of continuous functions is key to their usefulness and importance. In particular, when two Chu spaces represent objects of a common kind such as vector spaces, groups, manifolds, etc, the continuous functions between them turn out to be exactly the linear transformations, group homomorphisms, diffeomorphisms, or whatever the appropriate structure-preserving map is.