$C^1$ approximation of a continuous curve.

Suppose I have two points $\alpha,\beta \in \Bbb{R}^n$. Define $$ X=\{\gamma \in C^1([0,1] , \Bbb{R}^n),\ \gamma(0)=\alpha,\gamma(1)=\beta ,0 <|\gamma'|<K\}$$ parametrized curves joining $\alpha$ and $\beta$. If I have a sequence $(\gamma_n) \subset X$ such that the paths $\gamma_n$ are all contained in a compact set $K \subset \Bbb{R}^n$ then this sequence is equi-bounded and equi-continuous (in $C([0,1],\Bbb{R}^n$) and by Ascoli-Arzela theorem there is a path $\gamma :[0,1] \to \Bbb{R}^n$ such that $\gamma_n$ converges uniformly to $\gamma$. The limit $\gamma$ is continuous, but may not be of class $C^1$.

In my case I can prove that $\gamma([0,1])\subset [\alpha,\beta]$ (the line segment joining $\alpha$ and $\beta$). My question is:

Can I find a sequence of paths $\theta_n \in C^1([0,1],[\alpha,\beta])$ for which $|\theta_n'| \leq K$ and $(\theta_n)$ converges uniformly (or maybe pointwise) to $\gamma$?

My intuition says that if I can approximate $\gamma$ with a sequence of $C^1$ paths joining $\alpha$ and $\beta$ then it is sufficiently regular such that I can approximate $\gamma$ with $C^1$ curves with $\theta_n([0,1])\subset \gamma([0,1])$.

One thought was to use projections of $\gamma_n$ on the segment $[\alpha,\beta]$, but I think that in some cases the projection can destroy differentiability.

2022-07-25 20:42:49
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