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It is known that every function $f(x)$ defined on the interval $(-a,a)$ can be represented as the sum of an even function and an odd function. However

How do you prove that this representation is unique?

Thanks for your help.

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@Sharkos What is the added value of that edit? –  Lord_Farin May 30 '13 at 13:28
    
Just fixing a minor grammatical error, apologies if it's too trivial to warrant correcting. –  Sharkos May 30 '13 at 13:33
    
@Sharkos the sentence "How to prove A" is not grammatically correct? I'm not a native speaker, that's why I'm asking. –  Ilya May 30 '13 at 13:51
    
The problem is "How to prove A?" isn't really a proper phrasing of a question. The construction "how to X" forms a noun; you might say "I don't know [how to X]" for example. To turn it into a question, you use "Can you tell me [how to X]?" or "How do you X?" for example. You will sometimes hear people use it like a question in speech, but in the same way you might say "Sausages?" rather than "Are those sausages?" or "Do you want some sausages?" or "Will there be sausages?" It reads strangely to my eye, which is why I occasionally change it. I usually only do so as part of larger edits. –  Sharkos May 30 '13 at 13:59
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1 Answer

up vote 8 down vote accepted

If $f = g_1 + g_2 = h_1+h_2$ where $g_1,h_1$ are even and $g_2,h_2$ are odd then $$ g_1 - h_1 = g_2 - h_2 \tag{1} $$ where the left-hand side of $(1)$ is even and the right-hand side is odd, hence both sides are just $0$. Indeed, it is easy to show just from the definitions that any function which is both even and odd must be a constant $0$.

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What's meaning of LHS? And why both sides need be $0$? –  Paul May 30 '13 at 13:06
    
@Paul LHS = left-hand side. I've edited a bit –  Ilya May 30 '13 at 13:13
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