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Prove that f(x)gi v e nb yf(x+y)=f(x)+f(...

Prove that `f(x)gi v e nb yf(x+y)=f(x)+f(y)AAx in R` is an odd function.

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Given `f(x+y) =f(x)+f(y) AA x in R " (1) " `
Replacing y by -x, we have
`f(x-x) = f(x)+f(-x)`
or `f(x) +f(-x) =f(0) " (2) " `
Now, putting ` x=y=0` in (1), we have
`f(0+0)=f(0)+f(0)`
or `f(0)=0`
Then from (2), `f(x)+f(-x)=0`. Hence, f(x) is an odd function.
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