Definition of multiplicity of a root of a function It is well known that $\rho \in \mathbb{R}$ is a root of order m of a real polynomial $P$  if $(x-\rho)^m$ divides $P$ but $(x- \rho)^{m+1}$ does not divide $P$. The same holds for holomorphic functions. Also, we can define the multiplicity of a root of a differentiable real valued function of one real variable by saying that $\rho$ if a root of f of multiplicity m if $f(\rho)=...=f^{(m-1)}(\rho)=0$ and $f^{(m)}(\rho) \neq 0$ (correct me if I'm wrong). I was wondering if we can define this notion of multiplicity of a root for a more general class of functions that generalizes the definition given above (let's say for example continuous or even discontinuous functions). If so how general can that class be and what would the definition look like?
 A: As a real function, $x^{4/3}$ is differentiable once at $0$ but $0$ doesn't have integral multiplicity. Similarly, for a function like $e^{-1/x^2}$ we get an isolated zero of infinite multiplicity for a $C^{\infty}$ function so these examples should show that you need a continuous extension to a small neighborhood in the complex plane to discuss about integral multiplicity (extension which is automatic in the polynomial case or more generally for a real analytic function)
in general, one can define the multiplicity at $w$ for a continuous function $f$ which is non-zero on a punctured disc around $w$;
namely, we use a standard lemma - if an (oriented) plane loop $\gamma$ doesn't pass through the origin, every parametrization $z=z(t), a \le t \le b$ of it, can be written as $e^{\alpha(t)}$ with continuous $\alpha(t)$ and then $\frac{\alpha(b)-\alpha(a)}{2\pi i}$ is an integer that depends only on $\gamma$ and is called as usual $n(\gamma, 0)$ the winding number of $\gamma$ wr $0$; then if $f$ is continuous and non-zero on a loop $\gamma$, we define $d(f,\gamma)=n(f \circ \gamma , 0)$
In particular, if $f$ is continuous and non-zero in a punctured disc $0<|z-w|<R$ centered at $w$, letting $C_r$ the positively oriented circle centered at $w$ and radius $r$ we have a well defined integer $d(f,C_r)$ for any $0<r<R$ and by usual homotopy properties the integer doesn't depend on $r$ and is called the multiplicty of $f$ at $w$, or $m(f,w)$
It doesn't matter if the function is defined (continuously), finite, etc at $w$ - the multiplicity will be an integer and will coincide with the usual one for a meromorphic function with a zero/pole at $w$ and will be zero if the function is continuous and non-zero at $w$.
However one can have cases (with very regular functions - harmonic polynomials for example) of zeroes with negative ($\bar z$ has multiplicity $-1$ at the origin - more generally anti-analytic functions have negative multiplicity at isolated zeroes) or even zero multiplicity ($iz^2+2\Re z$ has multiplicity zero at the origin), or cases where the multiplicity is $1$ but the function is not locally injective as in the analytic case ($2\Re z -z^2$ has multiplicity $1$ at the origin but is not $1-1$ on any small disc around it), so one has to be really careful about how the usual holomorphic intuition translates
