Return to Answer

$\newcommand\ep\varepsilon$Take any $\ep\in(0,1/2)$. Let $$A:=A_0\cup A_1\cup F,$$ where $A_0:=[0,\ep)\times B$, $A_1:=(1-\ep,1]\times B$, $F:=[0,1]\times\{0\}$, and $B$ is the unit ball.

You have a well-defined function $f$ on $A$ such that $f|_{A_0}=\rho_0$, $f|_{A_1}=\rho_1$, and $f(\cdot,0)=F_0$.

Your question is then when $f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$.

For that it is clearly necessary that $\rho_0$ and $\rho_1$ be extendible to smooth functions $\bar\rho_0$ and $\bar\rho_1$ on the closures $\bar A_0$ and $\bar A_1$ of $A_0$ and $A_1$. Then $f$ can be accordingly extended to a uniquely determined function $\bar f$ on the closure $\bar A$ of $A$.

So, the question then becomes when $\bar f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$.

By Whitney's Theorem I, a sufficient condition for this is that $\bar f$ be of class $C^\infty$ on $\bar A$ in the sense of this paper by Whitney (the set $A$ in the mentioned theorem can be any closed subset of a Euclidean space -- it does not have to be a manifold).

It is clear that this sufficient condition is also necessary.

Therefore and because the functions $\bar\rho_0$ and $\bar\rho_1$ are smooth, we conclude:

$f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$ iff condition (3.2) in the mentioned paper by Whitney holds for the function $\bar f$ and all the points $x^0$ in the set $[\ep,1-\ep]\times\{0\}$ (with respect to the set $\bar A$).

$\newcommand\ep\varepsilon$Take any $\ep\in(0,1/2)$. Let $$A:=A_0\cup A_1\cup F,$$ where $A_0:=[0,\ep)\times B$, $A_1:=(1-\ep,1]\times B$, $F:=[0,1]\times\{0\}$, and $B$ is the unit ball (you did not say if your unit ball is closed or open; let us assume that it is closed).

You have a well-defined function $f$ on $A$ such that $f|_{A_0}=\rho_0$, $f|_{A_1}=\rho_1$, and $f(\cdot,0)=F_0$.

Your question is then when $f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$.

For that it is clearly necessary that $\rho_0$ and $\rho_1$ be extendible to smooth functions $\bar\rho_0$ and $\bar\rho_1$ on the closures $\bar A_0$ and $\bar A_1$ of $A_0$ and $A_1$. Then $f$ can be accordingly extended to a uniquely determined function $\bar f$ on the closure $\bar A$ of $A$.

So, the question then becomes when $\bar f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$.

By Whitney's Theorem I, a sufficient condition for this is that $\bar f$ be of class $C^\infty$ on $\bar A$ in the sense of this paper by Whitney (the set $A$ in the mentioned theorem can be any closed subset of a Euclidean space -- it does not have to be a manifold).

It is clear that this sufficient condition is also necessary.

Therefore and because the functions $\bar\rho_0$ and $\bar\rho_1$ are smooth, we conclude:

$f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$ iff condition (3.2) in the mentioned paper by Whitney holds for the function $\bar f$ and all the points $x^0$ in the set $[\ep,1-\ep]\times\{0\}$ (with respect to the set $\bar A$).

The case $\ep>1/2$ is trivial. In the case when $\ep=1/2$, the obvious necessary and sufficient condition is that the values of the function $\bar\rho_0$ and all its partial derivatives on the set $\bar A_0\cap\bar A_1=\{1/2\}\times B$ be the same as the corresponding values of the function $\bar\rho_1$ and all its partial derivatives.

$\newcommand\ep\varepsilon$Without loss of generality, $\ep\in(0,1/2]$. Let $$A:=A_0\cup A_1\cup F,$$ where $A_0:=[0,\ep)\times B$, $A_1:=(1-\ep,1]\times B$, $F:=[0,1]\times\{0\}$, and $B$ is the unit ball.

You have a well-defined function $f$ on $A$ such that $f|_{A_0}=\rho_0$, $f|_{A_1}=\rho_1$, and $f(\cdot,0)=F_0$.

Your question is then when $f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$.

For that it is clearly necessary that $\rho_0$ and $\rho_1$ be extendible to smooth functions on the closures $\bar A_0$ and $\bar A_1$ of $A_0$ and $A_1$. Then $f$ can be accordingly extended to a uniquely determined function $\bar f$ on the closure $\bar A$ of $A$.

So, the question then becomes when $\bar f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$.

By Whitney's Theorem I, a sufficient condition for this is that $\bar f$ be of class $C^\infty$ on $\bar A$ in the sense of this paper by Whitney (the set $A$ in the mentioned theorem can be any closed subset of a Euclidean space -- it does not have to be a manifold).

It is clear that this sufficient condition is also necessary.

Therefore and because the functions $\rho_0$, $\rho_1$, and $F_0$ are smooth, we conclude:

$f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$ iff condition (3.2) in the mentioned paper by Whitney holds for the function $\bar f$ and the points $x^0=(\ep,0)$ and $x^0=(1-\ep,0)$ in the set $\bar A$ (with respect to the set $\bar A$).

$\newcommand\ep\varepsilon$Take any $\ep\in(0,1/2)$. Let $$A:=A_0\cup A_1\cup F,$$ where $A_0:=[0,\ep)\times B$, $A_1:=(1-\ep,1]\times B$, $F:=[0,1]\times\{0\}$, and $B$ is the unit ball.

You have a well-defined function $f$ on $A$ such that $f|_{A_0}=\rho_0$, $f|_{A_1}=\rho_1$, and $f(\cdot,0)=F_0$.

Your question is then when $f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$.

For that it is clearly necessary that $\rho_0$ and $\rho_1$ be extendible to smooth functions $\bar\rho_0$ and $\bar\rho_1$ on the closures $\bar A_0$ and $\bar A_1$ of $A_0$ and $A_1$. Then $f$ can be accordingly extended to a uniquely determined function $\bar f$ on the closure $\bar A$ of $A$.

So, the question then becomes when $\bar f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$.

By Whitney's Theorem I, a sufficient condition for this is that $\bar f$ be of class $C^\infty$ on $\bar A$ in the sense of this paper by Whitney (the set $A$ in the mentioned theorem can be any closed subset of a Euclidean space -- it does not have to be a manifold).

It is clear that this sufficient condition is also necessary.

Therefore and because the functions $\bar\rho_0$ and $\bar\rho_1$ are smooth, we conclude:

$f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$ iff condition (3.2) in the mentioned paper by Whitney holds for the function $\bar f$ and all the points $x^0$ in the set $[\ep,1-\ep]\times\{0\}$ (with respect to the set $\bar A$).

$\newcommand\ep\varepsilon$Without loss of generality, $\ep\in(0,1/2]$. Let $$A:=A_0\cup A_1\cup F,$$ where $A_0:=[0,\ep)\times B$, $A_1:=(1-\ep,1]\times B$, $F:=[0,1]\times\{0\}$, and $B$ is the unit ball.

You have a well-defined function $f$ on $A$ such that $f|_{A_0}=\rho_0$, $f|_{A_1}=\rho_1$, and $f(\cdot,0)=F_0$.

Your question is then when $f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$.

By Whitney's Theorem I, a sufficient condition for this is that $f$ be of class $C^\infty$ on $A$ in the sense of this paper by Whitney; the set $A$ in the mentioned theorem can be any closed subset of a Euclidean space -- it does not have to be a manifold.

It is clear that this sufficient condition is also necessary.

Therefore and because the functions $\rho_0$, $\rho_1$, and $F_0$ are smooth, we conclude:

$f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$ iff condition (3.2) in the mentioned paper by Whitney holds for the points $x^0=(\ep,0)$ and $x^0=(1-\ep,0)$ in the set $A$ (with respect to the set $A$).

$\newcommand\ep\varepsilon$Without loss of generality, $\ep\in(0,1/2]$. Let $$A:=A_0\cup A_1\cup F,$$ where $A_0:=[0,\ep)\times B$, $A_1:=(1-\ep,1]\times B$, $F:=[0,1]\times\{0\}$, and $B$ is the unit ball.

You have a well-defined function $f$ on $A$ such that $f|_{A_0}=\rho_0$, $f|_{A_1}=\rho_1$, and $f(\cdot,0)=F_0$.

Your question is then when $f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$.

For that it is clearly necessary that $\rho_0$ and $\rho_1$ be extendible to smooth functions on the closures $\bar A_0$ and $\bar A_1$ of $A_0$ and $A_1$. Then $f$ can be accordingly extended to a uniquely determined function $\bar f$ on the closure $\bar A$ of $A$.

So, the question then becomes when $\bar f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$.

By Whitney's Theorem I, a sufficient condition for this is that $\bar f$ be of class $C^\infty$ on $\bar A$ in the sense of this paper by Whitney (the set $A$ in the mentioned theorem can be any closed subset of a Euclidean space -- it does not have to be a manifold).

It is clear that this sufficient condition is also necessary.

Therefore and because the functions $\rho_0$, $\rho_1$, and $F_0$ are smooth, we conclude:

$f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$ iff condition (3.2) in the mentioned paper by Whitney holds for the function $\bar f$ and the points $x^0=(\ep,0)$ and $x^0=(1-\ep,0)$ in the set $\bar A$ (with respect to the set $\bar A$).