If in a neighborhood of $x_0$ we have $$f(y-y_0) = (x-x_0)+o(x-x_0),$$ where $f$ is a monotone continuous function. How to show there must be a continuous function $g$ such that $$(x-x_0) = g(y-y_0)$$ in a neighborhood of $y_0$?
In fact, I want a general existence theorem for inverse functions by Taylor expansion, but I am not sure the above assert.
A function has an inverse in a neighborhood of a point $y_0$ iff this function is bijective in a neighborhood of $y_0$, that means iff this function is surjective and injective in a neighborhood of $y_0$.
A function is representable in a neighborhood of $y_0$ as a Taylor series, iff this function is differentiable in a neighborhood of $y_0$. If a function is differentiable in a neighborhood of $y_0$, this function is continuous in a neighborhood of $y_0$.
The general existence theorem for inverses by Taylor expansion that you are looking for is the Lagrange inversion theorem.
The theorem states that a function $f\colon x\mapsto y=f(x)$, analytic at a point $x_0$, with $f'(x_0)\neq 0$, has an inverse in a neighborhood of $y_0=f(x_0)$ that is a convergent power series, means an analytic function, and gives a formula for the general term of that power series.