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Let $a\le b$ and $k\ge 0$ be given and fixed. Let furthermore $x$ and $y$ denote two different elements of a Hilbert space $H$. Suppose $u:\mathbb{R}\rightarrow H$ is a $C^k$-curve in a Hilbert space C^k$-embedding connecting$H$with non-vanishing derivative x$ and $y$, s.t. the derivatives up to order $k$ vanish at infinity and $f:\mathbb{R} \rightarrow \mathbb{R}$ a given $C^k$-map with support in $[a,b].$ Then it it possible to construct a $C^k$map $\tilde{f}:H\rightarrow \mathbb{R}$ satisfying $\tilde{f}(u(s))=f(s)$ for any $s\in \mathbb{R}.$ (This can be done using a tubular neighborhood and a suitable cutoff function.)
Is it possible to make the map $f\mapsto\tilde{f}$ continuous? The domain of this map should be the space of $C^k-$curves which are embeddings on $[a,b]$ C^k$-embeddings (as above) times$\mathbb{R}$-valued$C^k$- functions supported in$[a,b]$. And the codomain should be the space of$C^k$-maps on$H$. 3 edited body Let$a\ge a\le b$and$k\ge 0$be given and fixed. Suppose$u:\mathbb{R}\rightarrow H$is a$C^k$-curve in a Hilbert space$H$with non-vanishing derivative and$f:\mathbb{R} \rightarrow \mathbb{R}$a given$C^k$-map with support in$[a,b].$Then it it possible to construct a$C^k$map$\tilde{f}:H\rightarrow \mathbb{R}$satisfying$\tilde{f}(u(s))=f(s)$for any$s\in \mathbb{R}.$Is it possible to make the map$f\mapsto\tilde{f}$continuous? The domain of this map should be the space of$C^k-$curves which are embeddings on$[a,b]$times$\mathbb{R}$-valued$C^k$- functions supported in$[a,b]$. And the codomain should be the space of$C^k$-maps on$H$. 2 added 4 characters in body Let$a\ge b$and$k\ge 0$be given and fixed. Suppose$u:\mathbb{R}\rightarrow H$is a$C^k$-curve in a Hilbert space$H$with non-vanishing derivative and$f:\mathbb{R} \rightarrow \mathbb{R}$a given$C^k$-map with support in$[a,b].$Then it it possible to construct a$C^k$map$\tilde{f}:H\rightarrow \mathbb{R}$satisfying$\tilde{f}(u(s))=f(s)$for any$s\in \mathbb{R}.$Is it possible to make the map$f\mapsto\tilde{f}$continuous? The domain of this map should be the space of$C^k-$curves which are embeddings on$[a,b]$times$\mathbb{R}$-valued$C^k$- functions supported in$[a,b]$. And the codomain should be tha the space of$C^k$-maps on$H\$.