The paths of the roots of the quadratic function - mold, model and style

Abstract

By fixing two real coefficients and varying the remaining real
coefficient of the quadratic function, the model of the paths
described by the two roots of the function in the complex plane
is obtained (Yamaoka (2023)). A style is a set of models that
are related to each other and share the same mold -structure-
(except the style 4 originating from 𝑓𝑎 (𝑧) = 𝑎𝑧^2, 𝑎 ∈ R∗, consisting
of a single model). We determine the number of connected
components of the path of each root per model. We determine
the connected components of each mold. We give examples of
the molds. We discuss the continuity and differentiability of the
two roots: they are continuous in their domains and the simple
roots are infinitely differentiable (with the exception of the infinitely
differentiable zero double root of 𝑓𝑎 (𝑧) = 𝑎𝑧^2, 𝑎 ∈ R∗,
the other double roots that appear in the text are not differentiable).
The theoretical foundation that supports the results
obtained here belongs to Classical Analysis.