Products of function equivalence classes

[The1TL]

Homework Statement

If f ∈ C(R) with f(0) ≠ 0, show that there exisits a g ∈ C(R) such that [fg] = [1], where [1] denotes the equivalence class containing the constant function 1.

Homework Equations

The Attempt at a Solution

Let f ∈ C(R) such that f:R → R is defined as f(x) = 1/x and let g ∈ C(R) such that g:R → R is defined as g(x) = x. Therefore [fg] = [x/x] = [1] for all x∈R.

is this correct?

 

[HallsofIvy]

"The equivalence class containing the constant function 1" with what equivalence relation?