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Suppose I have an odd, increasing function $h$ with $h(0)=0$ and an unknown increasing function $f(D)$, $f(0)=0$.

Let:

\phi(h(f(D)) h(f(D)) h'(f(D)) f'(D)=D

where $\phi$ is the standard normal pdf

From the above equation, we can see that $D$ spans the real line and the when the RHS is $> 0$ so is the LHS and vice versa.

But, I can rewrite the above equation as:

$\frac{-\partial\phi(h(f(D))}{\partial{D}}=D$

$\implies 0.5 D^2=-\phi(h(f(D)) $

which does not make sense since the LHS is $> 0$ and the RHS is $< 0$

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    ^ Yes. (Well, nonnegative technically.)2012-04-10

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