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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.

  • 0
    Can you clarify what it means for the graph of two functions $[0, 1] \to \mathbb{R}^2$ to intersect?2012-01-25
  • 0
    With "graph" I meant the subset of $\mathrm R^3$, that is, for every $f \in \mathcal F$, there is $t \in [0,1]$ such that $f(t) = g(t)$.2012-01-25

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