Let be any well-defined function.

We want to express it as a sum of an even function and an odd function.

Let us define two other functions as follows:

and .

Claim I: F(x) is an even function.

Proof I; Since by definition , so so that F(x) is indeed an even function.

Claim 2: G(x) is an odd function.

Proof 2: Since by definition , so so that G(x) is indeed an odd function.

Claim 3:

Proof 3: indeed.

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