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A few more answers:

$h$ can be made to be a unit (=nowhere zero) in a small enough neighborhood; this is just manipulating the Taylor series in local coordinates.

The set of "simple" (=non-ramified) points is open and dense (think in terms of derivatives). If you are interested in the image of the set of ramified points under $f$, Sard's theorem can help.

Regarding small perturbations, "splitting" a ramified point into simple points is not the only possible scenario. Compare two perturbations of $f(z)=z^3$ in $\mathbb{C}$: $z^3+az^2$ and $z^3+a$, $z$ and $a$ in a neighborhood of $0$. I am not sure what you mean by "large" perturbations.

An alternative intuition for Riemann-Hurwitz on compact surfaces if the critical points of the map $f$ are nondegenerate can be given using a Morse function on $X'$; see

MR2126710 (2006a:30040) Stawiska, Małgorzata: Riemann-Hurwitz formula and Morse theory. The $p$p-harmonic equation and recent advances in analysis, 209–211, Contemp. Math., 370, Amer. Math. Soc., Providence, RI, 2005

2 added 26 characters in body

A few more answers:

$h$ can be made to be a unit (=nowhere zero) in a small enough neighborhood; this is just manipulating the Taylor series in local coordinates.

The set of "simple" (=non-ramified) points is open and dense (think in terms of derivatives). Sard's theorem can help.

Regarding small perturbations, "splitting" a ramified point into simple points is not the only possible scenario. Compare $z^3+az^2$ and $z^3+a$, $z$ and $a$ in a neighborhood of $0$. I am not sure what you mean by "large" perturbations.

An alternative intuition for Riemann-Hurwitz on compact surfaces if the critical points of the map $f$ are nondegenerate can be given using a Morse function on $X'$; see

MR2126710 (2006a:30040) Stawiska, Małgorzata: Riemann-Hurwitz formula and Morse theory. The $p$p-harmonic equation and recent advances in analysis, 209–211, Contemp. Math., 370, Amer. Math. Soc., Providence, RI, 2005

1

A few more answers:

$h$ can be made to be a unit (=nowhere zero) in a small enough neighborhood; this is just manipulating the Taylor series in local coordinates.

The set of "simple" (=non-ramified) points is open and dense (think in terms of derivatives).

Regarding small perturbations, "splitting" a ramified point into simple points is not the only possible scenario. Compare $z^3+az^2$ and $z^3+a$, $z$ and $a$ in a neighborhood of $0$. I am not sure what you mean by "large" perturbations.

An alternative intuition for Riemann-Hurwitz on compact surfaces if the critical points of the map $f$ are nondegenerate can be given using a Morse function on $X'$; see

MR2126710 (2006a:30040) Stawiska, Małgorzata: Riemann-Hurwitz formula and Morse theory. The $p$p-harmonic equation and recent advances in analysis, 209–211, Contemp. Math., 370, Amer. Math. Soc., Providence, RI, 2005