Zero function question

Question

"The function $ g : \mathbb{R}$ $\rightarrow$ $\mathbb{R}$ satisﬁes $ g(ab) = ag(b) + bg(a)$"

For this equation, am I right in assuming that the function , $g$ , is a zero function?

I'm trying to prove that any input of g would equal zero, say $g(1) = 0$. I can only see this working if I assume, or prove that $g$ is a zero function.

proof by contradiction, perhaps? (I'd try to work something out but I'm rather short on time right now) — Maroon, Feb 11 '14 at 07:58
In these two cases, g(0) would equal 0 and g(1) would equal g(1) + g(1) which I'm tyring to get g(1)=0 — Benji_Bombadill, Feb 11 '14 at 08:06
If you have $g(1)=g(1)+g(1)$, substract $g(1)$ on both sides to get $0=g(1)$. — Christoph, Feb 11 '14 at 08:07
Well, from the equation $g(1)=g(1)+g(1)$, can you calculate what $g(1)$ is? — 5xum, Feb 11 '14 at 08:07
Oh shoot, g(1) would equal zero if you subtracted g(1) from both sides, yea? — Benji_Bombadill, Feb 11 '14 at 08:10

score 1 · Answer 1 · answered Feb 11 '14 at 08:11

1

Note that $$ g(1) = g(1\cdot 1) = 1\cdot g(1)+1\cdot g(1)=2\cdot g(1) $$ so $g(1)=0$. Moreover, $$ g(0)=g(0\cdot 0)=0\cdot g(0)+0\cdot g(0)=0. $$

answered Feb 11 '14 at 08:11

Brian Fitzpatrick

26,559

Yea, Chris and 5xum got there, thanks. – Benji_Bombadill Feb 11 '14 at 08:12

Zero function question

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