Is a smooth curve locally a graph?

Question

Let me first introduce a defitinition.

Definition (Smooth Curve) Let $\gamma\colon [a,b] \to \mathbb{R}^2$ a curve (continous function). We call $\gamma$ a $C^k$-smooth curve if $\gamma'(t) \neq 0$ for all $t \in [a,b]$ and if it is of class $C^k$.

So if $\gamma$ is $C^k$-smooth with $k > 0$, using this definition, does there exist for each point $v \in \gamma$ on the curve a coordinate system for the plane in which some piece of the curve containing $v$ is the graph $y = f(x)$ of a continuous function? If yes, what should a lowest $k$ can be? I'm guessing $k=1$.

I've tried thinking/finding some counter-examples, but all the counter-examples I find aren't smooth (see simple closed curves, this post asks the same question but for simple closed curves). Also what is interesting about the post, was a comment by Daniel Fischer. He claimed that any simple closed curve with a convex interior is also locally a graph.

If there is not true, what extra conditions should I add for it be true?

Any help would be greatly appreciated. Thanks in advance!

Answer 1

Suppose you could parametrize your curve as $f(x,y)=0$. This could be very difficult to achieve explicitly, but once done, by the Implicit Function Theorem the assertion is valid for every point where $\frac{\partial f}{\partial y} \neq 0$.

Answer 2

I'm assuming that your curve has no self-intersection; otherwise it clearly won't look like like a graph near such an intersection (except it if it a closed cycle, which is also uninteresting).

For $v=\gamma(t_0)$, an appropriate rotation of your coordinate system will make $\gamma'(t_0)=(a,0)$ for some $a>0$.

However, then $\gamma_x$, the $x$-coordinate of $\gamma$, is a $\mathcal C^1$ function with positive derivative at $t_0$. So $\gamma_x$ has positive derivative in a neighborhood of $t_0$, and will therefore be strictly increasing in this neighborhood; and a strictly increasing continuous function has an inverse.

In this neighborhood, therefore, it is the graph of the function $x\mapsto \gamma_y(\gamma_x^{-1}(x))$.