If $dy/dx = 0$ for all $x$ in the domain, is $dx/dy$ also zero? This seems problematic because $dy/dx$ can be thought as $0/1 = 0$ but when you reverse the upper and lower part of the fraction, the fraction is an invalid number.
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If $dy/dx=0$, then $dx/dy$ is infinite. One way to visualize this is that an inverse function $x(y)$ may be seen as a $90^{\circ}$ rotation of the original function $y(x)$. In that case, a horizontal slope becomes a vertical one by inversion. |
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If $\frac{dy}{dx}=0$ in an open interval, then $y$ is constant in that interval, so one does not have an inverse function $x$ of $y$. |
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Note that $y$ is a function of $x$ and as $\displaystyle\frac{dy}{dx} = 0$, $y$ is a locally constant function. Furthermore as $\displaystyle\frac{dy}{dx}$ exists for every point in the domain, the domain must be open. As such, $y$ is not an injective function of $x$, so it can't have an inverse; in particular, $x$ is not a function of $y$ so $\displaystyle\frac{dx}{dy}$ doesn't make sense in this context. |
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