Tool Like Argument Principle For Real-Valued Functions

Question

The Argument Principle gives a way of numerically counting the number of roots-poles ($Z-P$) of a meromorphic function in a contour. I was wondering, can the Argument Principle (or some other tool like it) be used to count $Z-P$ (or just $Z$) of a real -valued function $f$, that is neither holomorphic nor meromorphic, but does satisfy certain conditions (like being $p$-times differentiable, being continuous, etc.) The tool should only count $Z-P$ (or $Z$) within a shape defined in the real plane.

Answer 1

Sadly not... I suppose the reason is that there are "too many" smooth functions. For instance, suppose you know what $f$ is on $(-\infty,a)$ and $(a+\epsilon, \infty)$, then there actually exist infinitely many smooth functions that agree with $f$ on those intervals but are different. Whereas if you work with holomorphic functions, if $f=g$ on ANY collection of points that accumulates in your domain $\Omega$, then $f\equiv g$ in $\Omega$.