$C^1$ 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 ?