Let's assume $n \ge 2.$ Suppose $f:\Bbb R^n \times \Bbb R \to \Bbb R^{n+1}$ has the form$$f(x,t) = (\phi_t(x),t),$$where $\phi_{t_0}:\Bbb R^n \to \Bbb R^n$ is a continuous function for each fixed $t_0$. Assuming also that there exists a dense set $D\subset \Bbb R^n$ such that for each $x_0 \in D$, the map $f(x_0,\cdot)$, which is the restriction of $f$ to the line $\{x_0\}\times \Bbb R$, is uniformly continuous.
Is it true that $f$ must be continuous?
I suspect that the answer is negative, but I cannot think of a counterexample yet. Of course, here we are talking about the standard Euclidean topology only.