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When I was wondering about if Constant Functions were even or odd, I thought about the function:

f(x) = 0 , It's simultaneously odd and even,

f(3) = 0 , f(-3) = 0, f(1) = 0 , -f(1) = 0,

(It's Indeed a counter-example) I'm pretty sure I haven't solved the profound paradoxes of math. Is what I have concluded right?, Has anyone heard of it?

P.S ( I was taught it can't be both, so YEAH)

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marked as duplicate by Harald Hanche-Olsen, Davide Giraudo, Eric Naslund, Daniel Rust, Bruno Joyal Nov 30 '13 at 16:06

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3  
What about the statement "A nonzero function cannot be both odd and even"? –  doppz Nov 30 '13 at 13:57
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Its like saying "I have proved that $0$ is both positive and negative" :P –  Abishanka Saha Nov 30 '13 at 13:59
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Yes, we heard of it. And actually, that's the only function that is both odd and even. –  Cure Nov 30 '13 at 14:01
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And your point is? –  Carsten Schultz Nov 30 '13 at 14:04
    
I was taught It can't be both. Thanks to you guys I know the truth :) –  MohaGain Nov 30 '13 at 15:02

2 Answers 2

As Clive Newstead says here:

Unlike integers, "not even" does not mean the same thing as "odd". You'll find situations like this all over the place: always work from the definitions, don't let your knowledge of the English language deceive you.

His advice goes for all mathematical concepts.

From the very definitions, there's no reason why a function can't be both odd and even. Recall that a function $f\colon\Bbb R\to \Bbb R$ is:

  1. Odd if, and only if, $\forall x\in \Bbb R(f(-x)=-f(x))$;
  2. Even if, and only if, $\forall x\in \Bbb R(f(-x)=f(x))$.

'Odd' and 'even' are just words to represent the above definitions. The meaning it has in the English language (or even in other mathematical concepts) has no bearing here.

You should also note that the concept of odd and even functions is only defined for symmetric domains, that is, domains such that if $x$ is the domain, so must $-x$ be too. See this link for further clarification.

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If a function $f\colon\mathbb{R}\to\mathbb{R}$ (the domain could also be a symmetric interval) is odd and even, you have, for all $x$ in the domain,

$$ f(x)=f(-x)=-f(-x) $$

that is,

$$f(x)=-f(x).$$

Can you draw a conclusion from this?

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