'Continuity' necessary in proof dynamical systems? The following is a (rough) translation of a statement and proof given during a course in dynamical systems.
Let $D \subseteq \mathbb{R}^2$ be and open set, $f: D \to \mathbb{R}: (t, x) \mapsto D(t,x)$ a continuous function such that $\frac{\partial f}{\partial x}$ (exists and) is continuous. Then $f$ is locally Lipschitz with respect to $x$.
Proof For $(\tau, \gamma) \in D$, since $D$ is open, pick a compact, convex neighbourhood $U$ of $(\tau, \gamma)$ such that $U \subseteq D$. Then $\frac{\partial f}{\partial x}(U)$ is compact, thus has some upper bound $C > 0$. Pick any $(t,x), (t, x') \in U$. By the Lagrange mean value theorem there exists some $x'' \in [x, x']$ such that
$$
\vert f(t,x) - f(t,x')\vert \leq \vert \frac{\partial f}{\partial x}(t,x'')\vert \vert x - x' \vert \leq C \vert x - x' \vert
$$
showing that $f$ is indeed Lipschitz with respect to $x$ after restricting to some neighbourhoud of $(\tau, \gamma)$. This by definition means that $f$ is locally Lipschitz with respect to $x$. (QED)
I might be mistaken - multivariate calculus was a few years ago - but surely the continuity of $f$ was not used (and thus not needed) in this proof? If that's true, why would my professor have included it?
 A: The continuity of $f$ is a consequence of the statement that $\frac{\partial f}{\partial x}$ (exists and) is continuous on $D$. For a function to be differentiable --to be more precise, for its derivative to exist / to be well defined-- everywhere in $D$, it is necessary for $f$ to be (at least!) continuous. See also the relevant Wikipedia lemma.
As you noted, the differentiability or continuity of $f$ in the $t$-direction is never used. However, the continuity of $f$ in $t$ follows from the larger context. That is, we want to make sure that the dynamical system
\begin{equation}
\frac{\text{d} x}{\text{d} t} = f(x,t)
\end{equation}
makes sense in some way. If $f$ is not continuous in $t$, this implies that $\frac{\text{d}x}{\text{d} t}$ is not continuous in $t$. This means that $x(t)$ is not continuously differentiable.
This does not immediately cause problems, but for a large number of results in dynamical systems theory, it is necessary to be able to assume that an orbit $x(t)$ is continuously differentiable in $t$. However, it is very well possible (and quite interesting, too) to relax the continuity assumption of $f$ in the $t$-direction, and see how previous results can be extended.
