I am very, very confused with the notion $\epsilon(f(x))s$.
To my understanding, $s$ is a map sends to $F(x,s)$, and $\epsilon$ is the distance function given a point $f(x)$. So what does $\epsilon(f(x))s$ means, when we just put them together? Multiplication of two functions?
Corollary. Let $f\colon X\to Y$ be a smooth map, $Y$ being boundaryless. Then there is an open ball $S$ in some Euclidean space and a smooth map $F\colon X\times S\to Y$ such that $F(x,0)=f(x)$, and for any fixed $x\in X$ the map $s\mapsto F(x,s)$ is a submersion $S\to Y$. In particular, both $F$ and $\partial F$ are submersions.
Proof. Let $S$ be the unit ball in $\mathbf R^M$, the ambient Euclidean space of $Y$, and define $$F(x,s)=\pi[f(x)+\epsilon(f(x))s].$$ Since $\pi\colon Y^\epsilon\to Y$ restricts to the identity on $Y$, $F(x,0) = f(x)$. For fixed $x$, $$s\mapsto f(x)+\epsilon(f(x))s.$$
Thanks for helping out!
Since we are living in the ambient Euclidean space of $Y$, say $E=\mathbb R^M$, points may be added componentwise like vectors, and multiplied by scalars similarly. Then since $f(x)\in Y\subset E$, and $s\in S\subset E$, and $\epsilon(f(x))\in \mathbb R$, it makes perfect sense to compute the scalar product of the second with the third, and then add the first. The result is some point in the ambient space $E$. This is then projected back into $Y$ using $\pi$.