How Do Operations on Germs of Functions at a Point Ensure Well-Definedness?

[The1TL]

Homework Statement

Let C(R) be th set of all continuous functions on ℝ and let O be the germs
of continuous functions at the origin. We have a natural surjective map π : C(R) → O. Define addition and multiplication in O by, [f ] ⊕ [g] = [f + g], [f ] ⊗ [g] = [f g].

Prove that the above operations are well defined

((This means, you must prove that if f ∼ F and g ∼ G, then f + g ∼ F + G and fg ∼ FG.))

Homework Equations

The Attempt at a Solution

I have no idea where to even start with this problem. My teacher tends to include problems on homework that do not pertain to his teachings or anything in our text. I have searched the internet to try to find more about how to solve this sort of problem but I cannot find anything. Any help is greatly appreciated.

 

[mighty2000]

To prove that the operations of addition and multiplication in O are well-defined, we need to show that they do not depend on the choice of representatives for the germs [f] and [g]. In other words, if we have two different representatives f and F for [f], and two different representatives g and G for [g], then [f + g] and [fg] will be the same regardless of which representatives we choose.

To prove this, we can use the fact that the map π : C(R) → O is surjective, meaning that every germ in O has at least one representative in C(R). So, let's assume that f and F are two different representatives for [f], and g and G are two different representatives for [g]. We want to show that [f + g] and [fg] are the same regardless of whether we use f and g or F and G as our representatives.

First, let's look at addition. Since f and F are two different representatives for [f], we know that f ∼ F. This means that f and F are equivalent at the origin, or in other words, they have the same value at x = 0. Similarly, since g and G are two different representatives for [g], we know that g ∼ G, meaning they have the same value at x = 0. Now, if we add f and g, we get the function f + g. Similarly, if we add F and G, we get the function F + G. Since f and F are equivalent at x = 0, and g and G are equivalent at x = 0, we know that f + g and F + G are also equivalent at x = 0. This means that [f + g] and [F + G] are the same germ, or in other words, they are equal in O. This shows that the operation of addition in O is well-defined.

Next, let's look at multiplication. Again, since f and F are two different representatives for [f], we know that f ∼ F. Similarly, g ∼ G. Now, if we multiply f and g, we get the function fg. If we multiply F and G, we get the function FG. Since f and F are equivalent at x = 0, and g and G are equivalent at x = 0, we know that fg and FG are also equivalent at x