Let f be a function such that f : U ⊆ R n → R...

antennense

antennense

Answered

2022-07-09

Let f be a function such that f : U R n R . U is open in R n and path connected and f is continuous. Let x 1 , x 2 U. Proof that for all c [ f ( x 1 ) , f ( x 2 ) ] there exists an x U such that f ( x ) = c. I'm supposed to use one-dimensional intermediate value theorem to proof this. There is a hint stating that I should look out for a function φ : [ 0 , 1 ] U such that we use a "useful" composition of f and φ. I really don't know how to do this proof I would appreciate help a lot!

Answer & Explanation

Alexis Fields

Alexis Fields

Expert

2022-07-10Added 14 answers

You could let ϕ be a continuous function satisfying ϕ ( 0 ) = x 1 and ϕ ( 1 ) = x 2 . Its existence is guaranteed by path-connectedness. The composition f ϕ is continuous, maps [ 0 , 1 ] to R , and satisfies f ϕ ( 0 ) = f ( x 1 ) and f ϕ ( 1 ) = f ( x 2 ). There must exist t [ 0 , 1 ] with f ϕ ( t ) = c: what can you say about ϕ ( t )?
Sonia Ayers

Sonia Ayers

Expert

2022-07-11Added 3 answers

you need to find a point x U satisfying f ( x ) = c. So far there is a point t [ 0 , 1 ], guess i`m right

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