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I need help solving this: if $f: \mathbb{R} \to \mathbb{R}$ is an odd convex function, then $f=ax $ for any a∈R

Daryl Joseph
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    Wouldn't it be concave as well? – Angina Seng Jun 16 '17 at 04:29
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    Note that $f(0)=0$ by the odd condition. Then think at a convex function that intersects a line at $\gt 2$ points, and why it can't be strictly convex. P.S. $f(x)=x$ is an odd convex non-constant function, so your question must be misstating something. – dxiv Jun 16 '17 at 04:33
  • So should the question be to prove thats its linear? – Daryl Joseph Jun 16 '17 at 11:20
  • Ok I've proven that if f is both convex and concave then this equality holds f(ax1+(a-1)x2)=af(x1)+(a-1)f(x2) ... but how can i prove that f=ax – Daryl Joseph Jun 24 '17 at 06:41

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The question is incorrect as pointed out by dxiv. $f(x)=x$ is a counter example.

I am addressing the question of what is an odd convex function.

Since $f$ is an odd function,

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

If $f(x)$ is convex, then $-f(x)$ is concave.

Hence $f(-x)$ is concave which imply that $f(x)$ is a concave function.

In summary $f(x)$ is both convex and concave. Can you conclude what type of function is both convex and concave?

Siong Thye Goh
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