If for all holds. Show f(x) is an odd function.

Obviously, the domain of f(x) is R. Let where then Let x = y = 0 where then Thus which means Hence f(x) is an odd function.

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[Solved] Proving the function

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June 15, 2021 10:44 am

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If for all $x,y \in \mathbb{R},$ $f(x+y)=f(x)+f(y)$ holds. Show f(x) is an odd function.

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June 15, 2021 10:44 am

Topic starter

Staff

(@armor)

Admin

Honorable Member

317 Posts
156 153 0

Obviously, the domain of f(x) is R.

Let $\small y=-x$ where $\small f(x+y)=f(x)+f(y),$ then $\small f(0)=f(x)+f(-x).$

Let x = y = 0 where $\small f(0)=f(0)+f(0),$ then $\small f(0)=0.$

Thus $\small f(x)+f(-x)=0$ which means $\small f(x)=-f(-x).$ Hence f(x) is an odd function.

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