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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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  .

Alexia Hart · Accepted Answer

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  ]

Brock Byrd · Accepted Answer

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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