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invioor 2022-07-11 Answered
Given continuous function g : [ a , b ] n R n R . By Weistress g has a max and a min.

Can I also conclude its image contains all values in-between this maximum and minimum?

I need this result to complete a proof but cannot seem to find a generalisation of the Intermediate Value Theorem to R n .
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Answers (2)

Alexia Hart
Answered 2022-07-12 Author has 19 answers
You can reduce it to the 1-dimensional case by connecting two points where the extrema are attained by a line.

Since the set is convex it contains that line:

set m = min x [ a , b ] n f ( x ), M = max x [ a , b ] n f ( x )
and choose x m such that f ( x m ) = m and x M such that f ( x M ) = M

f ( t ) := g ( t x m + ( 1 t ) x M ) is continuous from [ 0 , 1 ] [ m , M ]
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Brock Byrd
Answered 2022-07-13 Author has 2 answers
The generalisation that you are looking for is that every image of a connected set by a continous map is still connected, so your image is actually a connected set of R (i.e. an interval) which contains the max and the min of g, and so all intermediate values.
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