For example, a rational function is zero if and only if its numerator (which is a polynomial) is zero. Thus, a rational function which is not identically zero have only a finite number of roots.
Is the same conclusion valid for smooth algebraic functions? If so, what would a proof or a source?
Edit (in response to the comments). I'm particularly interested in a real-valued function of a real variable given explicitly by a formula obtained from the elementary algebraic operations (addition, subtraction, multiplication, division, roots).