Question #2382

If f from R to R is continuous and open, show that f is strictly monotone.

Expert's answer

If f is not monotone, then there are two distinct points a<b such that f(a)=f(b). Then by Weierstrass' theorem f has a maximum C and minimum c on the closed interval [a,b]. We can assume that at least one of the numbers f(C) or f(c), say f(C) differs from f(a)=f(b).

Then the image open interval

f( (a,b) ) = (f(a), C]

is not open, which contradicts to openness of f.

Then the image open interval

f( (a,b) ) = (f(a), C]

is not open, which contradicts to openness of f.

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