proving that if $f(x)$ odd then $f(0)=0$

Question

Need to prove that if $f(x)$ is an odd function that defined in the point: $x=0$,

So $f(0)=0$.

I know that odd function is: $f(-x)=-f(x)$

And that $f(x)=0$ is an odd function but dont know how to prove.

Thanks.

You say you did not know how to prove. Fine, you had probably never met it before. But did you try to work out for yourself how to prove it. What value stands out as being a good one to try in the only useful equation around: $f(-x)=-f(x)$? — almagest, Sep 13 '14 at 22:36
@almagest - i saw all the answers and still dont know how to prove. yes, i know that it has to be $0$ becuase $-0=0$ but still i dont see any "full" proof without explaninig the zero part. — dave, Sep 13 '14 at 22:39
There are two quite separate issues here. Understanding, and presenting a good proof. Both are important. But maybe that is wandering off topic. — almagest, Sep 13 '14 at 23:09

Sujaan Kunalan · Accepted Answer · 2014-09-13T22:37:52.433

2

For $x=0, f(0)=-f(-0)=-f(0)$, but this is saying that one number is equal to its negative. This is only true for $0$.

$a=-a\implies a=0$

edited Sep 13 '14 at 22:37

answered Sep 13 '14 at 22:31

Sujaan Kunalan

10,944

that i know but how do you prove that? – dave Sep 13 '14 at 22:33
That is the proof. We used the fact the $f(0)$ was odd, i.e, $f(0)=-f(-0)$. We then note that $-0=0$, so we can write $-f(-0)$ as $-f(0)$. We then have that $f(0)=-f(0)$. We conclude that $f(0)=0$. – Sujaan Kunalan Sep 13 '14 at 22:45
1

@dave: If you are asking how to prove that $f(0)=-f(0)$, let $a=f(0)$, and so $-a=-f(0).$
$a=-a\implies 2a=0\implies a=0$.
– Sujaan Kunalan Sep 13 '14 at 22:48
now i got all the steps thnks a lot! – dave Sep 13 '14 at 22:50

score 0 · Answer 2 · answered Sep 13 '14 at 22:26

0

Take that definition you have for an odd function and use the fact that -0=0, then treat f(0) as a variable to solve for like an algebra equation.

answered Sep 13 '14 at 22:26

Alan

16,582

can you show how? – dave Sep 13 '14 at 22:35

score 0 · Answer 3 · answered Sep 13 '14 at 22:36

$$\text {Applying the definition: }\ f(0)=-f(0) ,$$ $$\text {so if}\ f(0) =c≠0 \text{ you would have two different values in x=0, c and -c}$$ $$\text {which is clearly impossible, so you must have }\ f(0)=0$$ $$\text{because for every x in the domain you must have only one f(x)}$$ $$\text{and 0 is the only value that satisfy that relation.}$$

score -1 · Answer 4 · answered Dec 25 '17 at 16:05

-1

For a odd function,f(-x)=-f(x) For a given value 0 in domain, Put x=0, f(0)=-f(0) 2f(0)=0 f(0)= 0

answered Dec 25 '17 at 16:05

Lucky

1

proving that if $f(x)$ odd then $f(0)=0$

4 Answers4