Consider the Cantor set $\Delta\subset [0,1]$, and let $f\colon \Delta\to [0,\infty)$ be a continuous injection. Must $f$ be monotone on some uncountable closed subset of $\Delta$? Note that that Van Der Waerden's construction is not applicable here.
Functions from the Cantor set
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real-analysis
continuity
cantor-set