# Given continuous function g : [ a , b ] n </msup> &#x2282;<!-

Given continuous function $g:\left[a,b{\right]}^{n}\subset {\mathbb{R}}^{n}\to \mathbb{R}$. 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 ${\mathbb{R}}^{n}$.
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Alexia Hart
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 $m=\underset{x\in \left[a,b{\right]}^{n}}{min}f\left(x\right)$, $M=\underset{x\in \left[a,b{\right]}^{n}}{max}f\left(x\right)$
and choose ${x}_{m}$ such that $f\left({x}_{m}\right)=m$ and ${x}_{M}$ such that $f\left({x}_{M}\right)=M$

$f\left(t\right):=g\left(t{x}_{m}+\left(1-t\right){x}_{M}\right)$ is continuous from $\left[0,1\right]\to \left[m,M\right]$
###### Did you like this example?
Brock Byrd
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 $\mathbb{R}$ (i.e. an interval) which contains the max and the min of $g$, and so all intermediate values.