Let $x(t) : \mathbb{R}\rightarrow \mathbb{R}^n$ and $u(t) : \mathbb{R}\rightarrow \mathbb{R}^p$ be vector fields, where $u(t)$ is continuous and bounded. The field $x(t)$ is subject to the dynamics $$\dot{x} = f\left(x,u(t)\right)$$ $$x(0) = x_0$$
Suppose we have an analogous process, $\tilde{x}(t)$, such that $$\dot{\tilde{x}} = f\left(\tilde{x},\tilde{u}(t)\right)$$ $$\tilde{x}(0) = x_0$$ where $\tilde{u}(t)$ is a smooth function approximating $u(t)$. By the Weierstrass approximation theorem, $$(\forall \;\epsilon > 0)\;\exists\;\;\text{smooth}\;\tilde{u}\;\;\text{s.t.}\;\left\|\tilde{u}-u\right\|_\infty < \epsilon\hspace{50pt}(1)$$ where $\left\|\cdot\right\|_\infty$ refers to the $L^\infty$ norm.
My understanding and intuition is that if the function $f$ is sufficiently "well-behaved", we can achieve arbitrarily little difference between $x(T)$ and $\tilde{x}(T)$ for some finite $T > 0$ provided we have arbitrarily little difference between $u$ and $\tilde{u}$. That is, $$(\forall\;\delta > 0)\;\exists\;\epsilon > 0\;\;\text{s.t.}\;\left((\forall \;u,\tilde{u})\;\;\left\|\tilde{u}-u\right\|_\infty < \epsilon \implies \left\|\tilde{x}(T)-x(T)\right\|_{(\infty)} < \delta\right)\hspace{50pt}(2)$$
where $\left\|\cdot\right\|_{(\infty)}$ denotes the vector $\infty$-norm.
Combining Eq. (1) and Eq. (2):
$$(\forall\;\delta > 0,u)\;\exists\;\; \text{smooth}\; \tilde{u}\;\text{s.t.}\;\left\|\tilde{x}(T)-x(T)\right\|_\infty < \delta\hspace{50pt}(3)$$
In literal terms, it's always possible to find a smooth $\tilde{u}$ that approximates continuous function $u$ closely enough to match the final state $\tilde{x}(T)$ to $x(T)$ to arbitrarily high precision.
My questions:
Knowing nothing about $u$ except the fact that it's continuous, what are the necessary conditions on $f$ to be able to conclude Eq. (2) holds? I happen to know that $f$ is globally Lipschitz continuous in both $x$ and $u$. Is this sufficient?
Is there an accepted name or terminology for the type of continuity in Eq. (2), where the "squeezing together" of two functions in the functional-norm sense implies the convergence of dynamical processes driven by those functions? If there happens to be an accepted name/terminology for Eq. (3), I'd welcome that too.
How would one state Eq. (1) literally? For example, would it be correct to state that "the set of smooth functions is dense in the set of continuous functions"?