continuous, monotonic, bijective

[gusl]

There should be a single adjective to describe transformations f : R -> R that are continuous, monotonic, bijective. (sorta equivalent to the condition: f' is continuous, zero nowhere, and not converging to zero too quickly... I say "sorta" because f(x) = x^3 violates the "zero nowhere" condition)

(tangentially, I'm fairly confident that they form a group under composition, with the increasing functions as a subgroup)

Comments

[psifenix.livejournal.com]

(1) They do form a group, with the increasing functions being a subgroup of index 2 (and therefore a normal subgroup). More explicitly, you have a homomorphism from this class of functions into {+1,−1} under multiplication (obv. isomorphic to Z/2Z) where increasing functions are sent to +1 and decreasing functions are sent to −1; that is, the increasing functions comprise the kernel.

(2) Your derivative exists only piecewise. The condition you are looking for is not "zero nowhere" but "not identically zero on any open subset of R."

(3) Any two of the conditions {(i) strictly monotonic, (ii) continuous, and (iii) bijective} imply the third (for functions on R). So really you only need two adjectives, not three.

(4) A continuous bijection from R to itself has a continuous inverse, so these functions are the homeomorphisms of R with itself, or "topological automorphisms." In general, if f:X→Y is a continuous bijection it is called a "condensation." The inverse function of a condensation need not be continuous for arbitrary X,Y.

[gustavolacerda.livejournal.com]

thanks. "Condensation" sounds like a great word to put on my title. :-)

[psifenix.livejournal.com]

Sure! I like the idea of these homeomorphisms from R to itself though, because you can visualize how the function is working—dilating and contracting parts of the real line as though it were elastic.

[stepleton.livejournal.com]

"Condensation" sounds like a great word to put on my title. :-)

...as long as it's meaningful to most of your audience, and since you did not know the word before posting, are you certain that this is the case?

[gustavolacerda.livejournal.com]

Furthermore, they are closed under + ... but this can be encoded as composition with the increasing function λ(x).(x+k) . So composition is all we need.

My brain is really pushing this idea of a creating a CFG of condensations.
But given the above result, all derivation trees can normalized into single-branch derivations, i.e. sequences of functions... and we can use inverses too.

So here's a way to generating set: < oddlySymmetrizedPower, exp >.
oddlySymmetrizedPower is just a way of making all positive powers look like the odd powers.

I can also restrict my attention to the positive condensations.