I originally posted this question on the Mathematics StackExchange and got told to consider putting it on here, on MathOverflow. I will word the question a bit differently:

Let $X$ be a compact $k$-dimensional submanifold of $\mathbb{R}^n$, $Y\subseteq\mathbb{R}^n$ and $f\colon X\to Y$ be measurable. What do we need from $n$, $k$ and $Y$ in order to be able to find $T>0$ and some bounded $g\in C([0,T]\times\mathbb{R}^n, \mathbb{R}^n)$, $g$ locally Lipschitz w.r.t the second argument, such that if $u_{(g,x_0)}$ is the unique solution to the initial value problem $$ \dot{x}(t) = g(t,x(t)),\; t\in [0,T],\quad x(0) = x_0$$ we have $f \approx \left(f_g\colon X\to Y, x_0\mapsto u_{(g,x_0)}(T)\right)$ in some reasonable way?

E.g., is the set $\{f_g\colon\text{$g$ as above}\}$ dense in $(C(X,\mathbb{R}^n),\Vert\cdot\Vert_\infty)$? Is the answer trivially negative? What about the relationship between $k$ and $n$? Can someone point me to where to look for answers to this question? Does this question have a name?

Many thanks in advance. (Note that I am an undergraduate, so please excuse me and inform me if this question or its formulation are not relevant to this StackExchange)