Are there classifiers with infinite number of classes? I am writing something about time series classification and wrote the following:

A classifier is a function that for given input $x_1, ..., x_n$ (in this case representing data stored in audio file) returns $y \in C = \{c_1, ..., c_k\}$, where $C$ is a finite set of classes (like $\{M, F\}$).

But then I realised that it might not be true. However, I can't think of any example of a classifier with an infinite possible number of classes. Do they exist? If so, what are some examples?

Classifying images to say which integer was depicted would be a classification problem with a countably infinite set of classes, but you would also need an infinitely large image as the input data.  It would also be a rather bad way of solving the problem to view it as single classification problem, rather than construct a solution by classifying the digits and constructing the answer from the component parts (if you did it that way, you might not need an infinite dataset to train it).
I doubt there is a practical case of an infinite set of classes as we have rather finite brains and live rather finite lives (in time and space), so we rarely encounter infinities in a meaningful way.
