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3
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Can there exist two functions $f,g: \mathbb R \to \mathbb R$ so that $f$ is continuous, $g$ is discontinuous, and their graphs $\Gamma_f, \Gamma_g \subseteq \mathbb R^2$ are related by an isometry? (I think you can assume the isometry is a rotation.) The graph of a continuous function must be path connected, so a natural intermediate question is, "can a discontinuous function have path connected graph?" |
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5
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I think the question has an easy answer, and it's essentially a 1st course in analysis type question that reduces pretty quickly to an intermediate value theorem application. Here's a more general statement. Let $f : \mathbb R \to \mathbb R$ be continuous and $h : \mathbb R^2 \to \mathbb R^2$ a homeomorphism. Then if $h(graph(f))$ is the graph of a function $g : \mathbb R \to \mathbb R$, then $g$ is continuous. Sketch: consider the function $P : \mathbb R \to \mathbb R$ given by $P(x) = \pi_1 \circ h(x,f(x))$, where $\pi_1(x,y)=x$. This is a continuous monotone function by the assumptions, so it is an open map by the intermediate value theorem, so it's inverse exists and is continuous. The claimed function $g$ is then $g(x)=\pi_2 h(P^{-1}(x), f(P^{-1}(x))$ where $\pi_2(x,y)=y$. The question gives me the feeling it's a homework problem, but I've never seen it before. This was solved in meta, with a correction by Anton. I wish I had known about this question yesterday -- I would have put it on my analysis class final exam! |
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4
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The only meaning of the statement I can see is, the (quite obvious) fact that a 90 degrees rotation of the graph of a continuous, increasing function $f$ gives the graph of a function, plus the jumps. Talking of rotating graphs, a more interesting fact is that a 45 degree rotation transforms 1-Lipschitz graphs into monotone increasing graphs, and conversely. As a consequence, the theorems of a.e. differentiability for Lipschitz functions, and for monotone functions, can be deduced from each other. |
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