Answer to Question #275512 in Real Analysis for Emmanuel

Question #275512

Is there a continuous function f:[0,1]~>[0,1] that is not constant in any nontrivial interval such that f^-1{0} is uncountable?


1
Expert's answer
2022-01-19T13:06:32-0500

Yes. Just take your preferred non-constant continuous function "g:[0,1]\\to[0,1]"

such that "g(0)=g(1)=1". For instance, you can take "g(x)=x(1-x)"

Now take any Cantor set "K\\subset [0,1]" and, for each interval "(a,b)" of "[0,1]-K"

for "x\\isin (a,b)" define f to be

"f(x)=g(x-a\/b-a)"

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