I'm trying to prove the following: given a family $\mathcal F$ of Lipschitz functions $f: [0,1] \rightarrow \mathrm R^2$, with a common Lipschitz constant, such that $\{f(0): f \in \mathcal F\}$ is bounded, there exists a continuous function $g: [0,1] \rightarrow \mathrm R^2$ whose graph intersects each of the graphs of the functions in $\mathcal F$.
Since there are at most $\mathfrak c$ functions in $\mathcal F$, we can conclude that there exists an almost everywhere continuous function $g$ that solves the problem (for example, defining $g$ in Cantor set and extending linearly). However, a continuous approximation of such a function may not solve the problem.
I would appreciate other suggestions to solve this problem.