I've encountered this assertion and I'm wondering how it is proved. (Here, a multiple point is defined as a point whose local ring is not a DVR, [EDIT] and a curve is a variety whose function field has transcendence degree 1 over the base field).
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(You should perhaps make explicit what is a curve for you...)
There is an open non-empty set of smooth points in a curve. The multiple points are therefore in the complement, and hence are finitely many.
An irreducible algebraic (or even algebraic with no multiple components) curve $\cal C$ has certainly a finite number of multiple points, since these are the points that $\cal C$ has in common with other curves.
On the other hand, it's easy to come up with real analytic curves with infinitely many multiple points.