Continuous functions can be distinguished by their values at say rational points of [0 1].
Germs of analytic functions can be distinguished by derivatives at a point. 
So in both cases we see countable sets of "parameters" for out set.

Consider set of germs of continuous functions.

**Question:** Is there a countable set of parameters such that different germs have different values of params ?

If yes, is there any "nice" set of parameters ? Or one may prove existence, but impossible to construct them "constructively / explicitly" ? 

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Related question: ["Nice" functions on germs of continuous functions][1].
My informal question is: "How to thing of germs of continuous functions/
what are "coordinates" on that space ? ".
Any informal comments are welcome. 

  [1]: https://mathoverflow.net/questions/184768/nice-functions-on-infinite-dimensional-space-of-germs-of-continuous-functions