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?

A:

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 $x=-y$.

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