avec_holl
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Homework Statement
13. (a) Prove that any function f with domain \mathbb{R} can be written f = O + E where E is even and O is odd.
(b) Prove that this way of writing f is unique.
Homework Equations
N/A
The Attempt at a Solution
(a) Suppose that f is some function with domain \mathbb{R}. This implies that for any arbitrary real number a, both f(a) and f(-a) exist. We can now define even and odd functions such that f(a) = O(a) + E(a) and f(-a) = E(a) - O(a). Since f can be represented by the sum of even and odd functions at a single point, we can define O and E point-wise such that their sum is always equal to f.
I'm not sure if this is a reasonable argument, but if it is, how can I make it rigorous?
(b) Suppose not, then f = O_1 + E_1 = O_2 + E_2. This implies that O_1 = (E_2 - E_1) + O_2. However, since O_1 is necessarily odd and E_2 - E_1 is necessarily even, we have that E_1 = E_2 and consequently O_1 = O_2. Hence, this way of writing f is unique.
I'm not sure if this is a reasonable argument either and any suggestions would be appreciated.