calculus - How does one interpret $dfrac{dx}{dy}$ for a function which isn't invertible?

I was just going through the proof of derivative of inverse functions.

The statement reads:

If $y= f(x)$ is a differentiable function of $x$ such that it's inverse $x=f^{-1}(y)$ exists, then $x$ is a differentiable function of $y$ and it's derivative is $\dfrac{dx}{dy} = \dfrac{1}{\frac{dy}{dx}}, \dfrac{dy}{dx}≠0$

Which naturally arises few questions, what if the inverse $y=f(x)$ doesn't exist?

My Questions :

1. Does $\dfrac{dx}{dy}$ still have a meaning?


2. If so then what would it mean geometrically?


3. Would $\dfrac{dx}{dy} = \dfrac{1}{\frac{dy}{dx}}$ still hold? And if it does, then why do we even need the invertible condition in the statement?