Does there exist a continuous function $f:[0,1]\to [0,1]$ for which the following holds? $$\forall k\in\mathbb{R}, 0<k<1 \: :\:\mathscr{S}_k= \{x\in[0,1]\mid f(x)=x+k\}\: \text{is an uncountable set}$$
2026-04-05 17:56:51.1775411811
Continuous function which has uncountable intersections with each parallel line to $y=x$
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