Lecture
In this section we shall become acquainted with some fundamental notions of singularity theory, which we shall use CONSTANTLY in what follows.
3.1. Germs
Most of the statements used in our course are
local in character, i.e., they hold only in some sufficiently
small neighborhood of the point under consideration, the exact «size» of which is not known (and, indeed, not needed). Such, for example, are the Morse lemma and
the division theorem. There is a generally accepted way to emphasize this
locality without mentioning each time the existence of a suitable
neighborhood. It consists in speaking not of the objects themselves
(functions, mappings, curves, etc.), but of their germs.
Let us give the definition of a germ for functions; for other objects
it is analogous.
DEFINITION 3.1. We call two functions f and g equivalent
at a point x0 if there exists a neighborhood of the point x0 on which they
coincide identically.
This equivalence relation obviously possesses the properties
of reflexivity, symmetry, and transitivity, and it partitions the set of functions into disjoint equivalence classes, which are called germs at the point x0. Obviously, every germ
(equivalence class) has infinitely many representatives.
The local properties of functions (mappings, curves, etc.) are
the properties of their germs, i.e., properties possessed by every representative of a given germ in a sufficiently small neighborhood of the point under consideration. In fact, the use of the term «germ» serves
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as a shorthand: when we say that the germ of the function f at
the point x0 possesses some property, this means that there exists a neighborhood of the point x0 in which the function f possesses this
property.
EXAMPLE 3.1. The germ of the function sin x at the point 0 has exactly one
zero, while the germ of the function sin(1/x) has infinitely many zeros. Another
example: it is obvious that the germ of any smooth function f at the point 0
can be represented in the form f(x) = g(x)cos x with a suitable smooth function
g. But for the function f itself such a representation may fail to hold:
it suffices to choose a function satisfying the condition f(π/2) ≠ 0.
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