The continuity restriction
Given continuous g:R -> R, it is even more popular to find a
continuous f:R -> R such that f(f(x)) = g(x). This is a lot harder
than the general case to do. This section is very small since I just started
investigating this. However, the anonymous mathematician who had the same
ideas as me didn't seem to have a clear-cut answer to this problem, making
it all the worthwhile to investigate. Then again, maybe I am just
rediscovering old results again, in which case I still don't care! The
f(f(x)) problem in the continuous case is absolutely fascinating, and much
more non-trivial than in the previous case.
The problem of finding continuous f is clearly the same as finding a regular
root f, only there are restrictions on what we should do with the classes! The
best route seems to be describing a function's continuity in terms of classes
or vice versa.
Let T be a topology and let g:T -> T be continuous. Define T/g = T/K(g), the
quotient space which identifies members of the same class. This topology is
called the class topology of the function g. If we refer to
a specific function in terms of its variable, we write T/[g(x)].
If T = R with the usual topology, then R/[ex] =
[0, 1) and R/[x + 1] = S1, for example, where = is
homeomorphism. The class topology of f(x) = 2x is not Hausdorff, however,
and in fact consists of two circles and a point, and the only open set
containing the point is the whole space. I thought for a moment once that
R/[sin x] = [0, 1), but fortunately for me I was wrong. This allows
my rough and unsturdy intuition that the "special" isolated classes of
a continuous function have something to do with "extra points" in the
class space, such as the extra point in R/[2x]. Moreover, in a continuous
function, classes seem to come in groups of classes that look the same
and are "near" each other, it seems to me that investigation of
classification of these groups of classes would be beneficial. For example,
we can say with loose notation that R/[2x] = (S11 U
S12) + 1, where 1 is the extra point, the only open
set of which contains the entire space. There are three groups of classes:
Those with elements above 0, which are all near each other and look alike,
those below 0, and the trivial point at 0. Each represents
S11, S12, and the extra point
respectively. (All of my ideas are very shaky in this paragraph, as is
probably apparent.)
The problem of finding continuous roots seems to be related to solving the
original problem given certain restrictions of the classes to be paired and
the relation of different pairs based on the class topology of the original
function. However, as of yet I am actually a pretty crappy topologist, and in
fact the only thing I ever learned about quotient spaces was by browsing books
at a bookstore. In spite of this I continue to struggle through the problem and
appreciate any contributions.
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