## $C^1$ curves in $\mathbb R^2$ with $C^1$ inverse

Let $$\gamma$$ be (the parametrization of) a closed $$C^1$$ curve in the plane $$\mathbb R^2$$, i.e. a $$C^1$$-map $$\gamma \colon S^1 \to \mathbb R^2$$.

Suppose that $$\gamma$$ is injective (no self intersections). From well known facts, the map $$\gamma$$ is an omeomorphism onto its image: in particular, the inverse map of $$\gamma$$ is continuous.

Q1. Is the inverse also of class $$C^1$$? In other words, is it true that every closed curve is diffeomorphic to the unit sphere?

I suspect that the answer is negative, so I formulate a second, more restrictive question in which I hope the answer is affirmative:

Q2. Suppose that the total curvature of $$\gamma$$ is finite. Is the inverse also of class $$C^1$$?

## Uniform inequality for a $C^1$ function

Let $$f(x,y)\in C^1([a,b]\times[c,d])$$ such that $$\exists \xi\in [c,d] : f(\xi,y)\neq 0, \forall y\in [c,d].$$

By the continuity of $$f$$, we have $$|f(\xi,\cdot)|\geq \min\limits_{[a,b]} |f(\xi,\cdot)|=\alpha_0>0. \;(1)$$

I want to know if we can prove the following result:

There exists open intervals $$I\subset [a,b], \; J \subset [c,d]$$, and $$C>0$$ such that $$\forall (x,y)\in I\times J: |f(x,y)|\geq C>0, \; (2).$$ If we use continuity of $$f(\cdot,y)$$ for each $$y\in [c,d]$$, by $$(1)$$ there exists an open interval $$I_y$$ containing $$\xi$$ such that $$\forall y\in [c,d], \forall x\in I_y: |f(x,y)|\geq \alpha_0>0.$$ But this is not what I’m looking for. I need two independent intervals such that $$(2)$$ holds.

If this is not true in general, can someone give a counter-example ?