# "Nice" functions on infinite-dimensional space of germs of continuous functions at a point

Consider set of all germs of continuous functions at some point.

Question: What are some functions ("any/nice/constructive/whatever") from this set to R (reals) ? (Except evalution at point and made of it).

Before trying to be more precise with "nice" let me add some examples of "nice" in similar setups.

Example 1. If one considers germs of analytical functions, then derivatives at point are those "nice" functions and moreover they are "coordinates" - i.e. if one knows all derivatives - one knows the germ itself and so any other function on germs can be expressed via them.

Examples 2a,b. If one considers not the germs, but continuous functions themselves on say [0 1], then evaluation functionls at points [0 1] are those "nice" functions. And again they are "coordinates". As another set of functionals one can take Fourier coefficints.

"Counter" Examples: Recent MO question shows that there are NO continuous functions on germs in certain topology on germs. Comments to unasnwered question on Math.SE tells something like if there would be something "nice", then everyone would know it.

Motivational Examples If I am not mistaking then, there are certain functions which are "nice".

1) Holder exponent. I.e. sup of h: |f(x+a)-f(x)| < c x^h (it depends only on germ, right ? or I missed something ?), hmmmmm this sup can be inifinity and I want functions to take values in reals, let us cook way round: if this sup is more than 1, let my function to be equal be to 1 by hands.

2) Fractal dimension of graph of function. I.e. Consider continuous function y=f(x). Consider it on the interval [0 1/n]. Calculate fractal dimension of graph of f(x). Take upper limit of those dimensions with $n-> \infty$. (It depends only on germ, right ? or I missed something ?)

Above is revised version of question. Let me preserve old version.

"Understanding an infinitesimal neighbourhood of a point in the realm of continuous functions"

For analytic functions, the set of all derivatives at a point, $f^{(n)}(a)$ for $n = 0,1,\ldots$, give a set "coordinates" on the space of all functions (germs). So $\operatorname{Spec}(C[[(x-a)]])$ is an "infinitesimal neighbourhood" of the point $x=a$.

Now consider not analytic, but continuous functions and a point $x=a$. Consider the germs of continuous functions at this point.

Question: How to understand the "algebra" of functionals on the space of germs of continuous functions?

Are there good "coordinates" (like the derivatives in the analytic case)? Are there some nice examples of such functionals? Can one express fractal dimensions or Hölder exponents via "coordinates/nice functions"?

Motivation: I was thinking about various fractal dimensions of graphs of functions (or of germs). There are many of them and it is difficult to understand the relation between them, it would be nice if they can be expressed via something canonical.

• Can you even construct any linear functional on germs of continuous functions (besides multiples of evaluation at the point) without the axiom of choice? Oct 18 '14 at 22:26
• I agree that $\mathrm{Spec}(\mathbb C\llbracket(a-x)\rrbracket)$ is an (I hesitate to say "the") infinitesimal neighborhood the point $x=a$. But it is not the neighborhood on which germs of analytic functions are defined, which is a little bigger. Indeed, an analytic function has a power series whose radius of convergence is positive; this happens iff the coefficients of the power series grow slower than some exponential. So I would say that $\mathbb C\llbracket(a-x)\rrbracket$ has a subalgebra consisting of those power series with positive radius of convergence, and the germ of a point ... Oct 19 '14 at 0:30
• ... with respect to analytic functions is $\mathrm{Spec}($this subring$)$. Oct 19 '14 at 0:31
• I think that your "examples" all fall into the following rough category of functions: define some quantitative property of a function using a limit, and if the limit doesn't exist, set the function to some arbitrary value (could be $\infty$, if that's allowed). You could simply declare this whole class of functions as 'nice' and be done with it. But I don't think these functions would qualify as 'coordinates' in any usual sense, though. Oct 19 '14 at 20:16
• Example: smooth - 0, non-smooth - 1. Another example: scaling degree, max k such that $\lim_{s\to 0} f(st) / s^k$ is finite. Oct 19 '14 at 21:39

First an easy observation: The closure of the ideal $\lbrace f\in C(\mathbb R): f=0 \text{ near } 0\rbrace$ in $C(\mathbb R)$ with respect to the compact open topology equals the ideal $\lbrace f\in C(\mathbb R): f(0)=0\rbrace$. Thus, if you want your functional on the space $C_0$ of germs of continuous functions to be continuous when composed with the quotient mapping $C(\mathbb R)\to C_0$, then it is $f\mapsto F(f(0))$ for some map $F$.
On the other hand, consider the modulus of continuity mapping $\rho(f)(|x|) = \max\lbrace |f(y)-f(0)|: |y|\le |x|\rbrace$ which induces a nonlinear map $\rho: C_0\to C_0$. So if you can find a "nice" functional on the set of germs of symmetric monotone continuous functions which vanish at 0, then you are done. For example, the $\sup$ of all $k\in \mathbb N$ with $\rho(f)(x)\ge C (\exp^k(x)-\exp^k(0)$ for some $C>0$, where $\exp^k = \exp\circ\dots\circ\exp$ (k-times), might be a candidate.
• Yes to both of your questions. But I believe that any nontrivial and computable nice functional better be continuous in the quotient topology; but then it can only factor over $f\mapsto f(0)$ Oct 20 '14 at 17:43