What is this notation regarding the Inverse Function Theorem?

Question

A definition in my notes state:

If $Df(a)$ is invertible (as a matrix), then $f$ is invertible on an open set $U$ containing $a$.
So given that $f(x,y) = (a,b)$ and there exists a $C^1$ local inverse near (x,y) with derivative $Df^{-1}(a,b) = (Df(x,y))^{-1}$.

My query is on the $C^1$. What does this mean in this context or any context? A lot of questions in my homework also go along the lines of "Show that $f$ has a local $C^1$ inverse near ...".

Answer 1

A function of class $C^n$ is a function that is $n$ times continuously differentiable. The inverse function theorem states that the inverse function is continuous and differentiable, so you can safely take one time derivative (it does not imply second derivative also exists) so it is of class $C^1$.

Answer 2

$C^n$ means $n$ times differentiable, and the $n$ th derivative is continuous.

i.e.

1. $C^0$ functions are continuous.
2. $C^1$ functions are differentiable, and the derivative is continuous.
3. $C^\infty$ functions are infinitely differentiable. These are referred to as "smooth".