Inverse function in $\mathbb{R}^n$ and bijectivity

Question

I have some questions on invertability of functions in $\mathbb{R}^n$.

1. If I want to show that a function $f(x, y, z)$ is not injective, is it enough for me to show that it's Jacobian $|D_f| = 0$ at some point $(x_1, y_1, z_1)$?

2. How should I thing about the level curve of $F^{-1}(k)$?

3. What is the (intuitive) meaning behind the scalar value of $|D_f|$? Why does it help understand if the function has an inverse?

4. Intuitively, why is the Inverse Function Theorem true? $D_{f^{-1}}(f(x)) = [D_f(x)]^{-1}$

5. In order to show that a function is not invertible in a neighbourhood of a point $P$, is it enough to show that its $|D_f(P)| \neq 0$?

Thank you in advance! I'm really struggling to understand inverse functions and the implicit function in $\mathbb{R}^n$ so any help that could lead me to understand it better would be appreciated.

Answer 1

1. No it's not. if you take the function $f(x)=x^3$ its derivative is zero at the origin yet it's injective.

2. "level sets" you mean. They're the set of points in the domain of the function at which the function evaluates to the number $k$. For instance if you have $f(x,y,z) = x^2 +y^2 +z^2$ then for each $k$, $f^-1(k)$ will define the set of points which constitude the surface of a sphere with radius $\sqrt(k)$.

3. It's because the derivative/Jacobian gives you the sense whether the function goes "up" or "down" or changes direction. Perhaps an example will be clear: Consider the function $x^2$. When $x<0$ the derivative is negative and the function is injective, if you graph the function you'll notice it is decreasing as it approaches zero from the left, but something happens when you approach zero: the derivative is zero and the graph of the function changes direction, it starts going up. Since it's going up then it means it will be taking values it had previously taken and hence is no longer injective. So the derivative can sense these changes in directions.

4. First understand it in 1-d. Plot a function $y$ vs $x$ like for example $y=x^2$ then take any point $x_0$ and observe its slope being $2x_0$. Now rotate your grid by 90 degrees to see (locally) the function $x(y)$ and observe its slope at the point $y_0=y(x_0)$. Notice its slope will be $\dfrac{1}{2x_0}$. Essentially it tells you the slope of the inverse function is the inverse of the slope of the original function which should be intuitively clear on the graph. Generalizing it to multivariable functions is more tricky but the essence is the same.

5. You mean invertible. Yes, showing that the Jacobian is nonzero immediately says that the function is locally invertible, it's what the IFT says.

You'll notice my examples are mostly 1-D not higher dimensional. The reason for that is that in higher dimensions things become more technical and you shouldn't expect all the intuition that you have in 1-d to generalize (for instance, if the Jacobian is nonzero at every point it doesn't mean the function is invertible, unlike the 1D case); however I mention them for simplicity and since knowing these basic things above serves as a good start in getting intuition for higher dimensions.