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Miklos Schweitzer 1980_6

Source: Construct irreducible functions

January 28, 2009
functionreal analysisreal analysis unsolved

Problem Statement

Let us call a continuous function f:[a,b]→R2  <spanclass=′latex−italic′>reducible</span> f : [a,b] \rightarrow \mathbb{R}^2 \;<span class='latex-italic'>reducible</span> if it has a double arc (that is, if there are a≤α<β≤γ<δ≤b a \leq \alpha < \beta \leq \gamma < \delta \leq b such that there exists a strictly monotone and continuous h:[α,β]→[γ,δ] h : [\alpha,\beta] \rightarrow [\gamma,\delta] for which f(t)\equal{}f(h(t)) is satisfied for every α≤t≤β \alpha \leq t \leq \beta); otherwise f f is irreducible. Construct irreducible f:[a,b]→R2 f : [a,b] \rightarrow \mathbb{R}^2 and g:[c,d]→R2 g : [c,d] \rightarrow \mathbb{R}^2 such that f([a,b])\equal{}g([c,d]) and (a) both f f and g g are rectifiable but their lengths are different; (b) f f is rectifiable but g g is not. A. Csaszar
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