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Finding the inverse of a function given that $f^{ -1}$ is continuous at a point and $f(x)>0$.

Suppose $f$ is a continuous function on an open interval $I$ and $f(x)>0$ for all $x$ in $I$. Let $a$ and $b$ be two points in $I$ and suppose that $f$ has a continuous inverse function in $I$ except at $a$ and at $b$. How would I go about finding that inverse function?

To define an inverse function, we must have for each $x$:
$$\tag{1}f(x)
eq 0\text{ and }f^{ -1}(f(x))=x$$
The problem is that it is unknown in advance whether $f$ satisfies (1). That’s why we must check that $f$ has the inverse at $a$ and $b$, that is:
$$\tag{2}f(a)
eq 0\text{ and }f^{ -1}(f(a))=a$$
$$\tag{3}f(b)
eq 0\text{ and }f^{ -1}(f(b))=b$$
If (2) holds, (1) holds, so we have an inverse function $f^{ -1}$. If (3) holds, the equation $f(x)=f(y)$ is solved by $x=y$ or $x=-y$.

1. Field of the Invention
This invention relates to an improved apparatus for the separation of solids from a liquid phase, and is more particularly related to an improved waste water separator including a unique vacuum induction auger apparatus for the separation of solids from a liquid phase in a manner which avoids the use of a screen basket or a mechanical separation method.
2. The Prior Art
Water blowers and pressure cleaners are commonly used to clean floors, furniture, clothing, and other floor-to-surface contact surfaces. These cleaners generally consist of a water source with a discharge fitting which provides a pressure and velocity for propelling a liquid spray onto

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END USER LICENSE AGREEMENT – ELEMENT3D LICENSE AGREEMENT

1.
BY USING THE ELEMENT3D LICENSE FILES OR THE ELEMENT3D3D FILES (e.g., the ELEMENT3D3DINFOMOD FILE)
OTHERWISE THAN AS PROVIDED FOR IN THIS AGREEMENT, YOU ARE ELECTING TO BE BOUND BY THE TERMS OF THIS AGREEMENT.
IF YOU DO NOT AGREE TO ALL THE TERMS OF THIS AGREEMENT, DO NOT USE THE ELEMENT3D LICENSE FILES OR THE ELEMENT3D3D FILES.
2.
IF YOU ARE THE DIRECT BENEFICIARY TO A REASONABLE VALUE OF A LICENSED PRODUCT, THIS LICENSE AGREEMENT APPLIES TO THAT PRODUCT.
3.
ELEMENT3D LICENSE FILES. YOU MAY DOWNLOAD A DEMO VERSION OF THE ELEMENT3D LICENSE FILES TO SEE THE FUNCTIONS OF THE ELEMENT3D LICENSE FILES. THE ELEMENT3D LICENSE FILES ARE PROVIDED FOR TESTING PURPOSES ONLY.
YOU SHOULD BACKUP ALL YOUR DATA IN THE EVENT THAT YOU ARE UNABLE TO ACCESS YOUR DATA.
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