Hey Gianfranco,
open your eyes! Catastrophe theory will not solve your problems with multivalued functions, nor will the use of the term "multivalued function".
I think I clarified in detail what terms are there and how to use them:
1. There is an analytic function (single valued)
2. If the function has only isolated singularities choose slices starting at the singularities for extending this function (uniquely/single valued) to the sliced complex plane.
3. Determine branches of the function by paths that cross the slices.
If your original analytic function \( f \) is not injective, you can invert it at a point with \( f'(x_0)\neq 0 \). That means there is a unique inversion \( f^{-1} \) in a neighborhood of \( (x_0,f(x_0)) \). If you look at the branches of \( f^{-1} \) you will find among them also the other values \( x \) for which \( f(x)=y \), i.e. for each \( x \) with \( f(x)=y \) there will be a branch \( {f^{-1}}_k \) such that \( {f^{-1}}_k(y)=x \). Or in other words for each \( x \) there is a \( k \) such that \( {f^{-1}}_k(f(x))=x \).
For example the function \( \sqrt{x} \) (sliced as usual) has the branches \( \sqrt{x}_k = (-1)^k \sqrt{x} \), i.e. at all only 2 branches that is \( \sqrt{x} \) and \( -\sqrt{x} \). And hence \( \sqrt{x^2} \) is not always equal to \( x \) but we have to choose the right branch for example \( -\sqrt{(-2)^2}=-2 \) and \( \sqrt{2^2}=2 \). In general we have merely \( \sqrt{x^2}=|x| \). And no multivalued function consideration will save you from considering this.
open your eyes! Catastrophe theory will not solve your problems with multivalued functions, nor will the use of the term "multivalued function".
I think I clarified in detail what terms are there and how to use them:
1. There is an analytic function (single valued)
2. If the function has only isolated singularities choose slices starting at the singularities for extending this function (uniquely/single valued) to the sliced complex plane.
3. Determine branches of the function by paths that cross the slices.
If your original analytic function \( f \) is not injective, you can invert it at a point with \( f'(x_0)\neq 0 \). That means there is a unique inversion \( f^{-1} \) in a neighborhood of \( (x_0,f(x_0)) \). If you look at the branches of \( f^{-1} \) you will find among them also the other values \( x \) for which \( f(x)=y \), i.e. for each \( x \) with \( f(x)=y \) there will be a branch \( {f^{-1}}_k \) such that \( {f^{-1}}_k(y)=x \). Or in other words for each \( x \) there is a \( k \) such that \( {f^{-1}}_k(f(x))=x \).
For example the function \( \sqrt{x} \) (sliced as usual) has the branches \( \sqrt{x}_k = (-1)^k \sqrt{x} \), i.e. at all only 2 branches that is \( \sqrt{x} \) and \( -\sqrt{x} \). And hence \( \sqrt{x^2} \) is not always equal to \( x \) but we have to choose the right branch for example \( -\sqrt{(-2)^2}=-2 \) and \( \sqrt{2^2}=2 \). In general we have merely \( \sqrt{x^2}=|x| \). And no multivalued function consideration will save you from considering this.