Generalisation of Intermediate Value Theorem

Question

Given continuous function $g:[a,b]^n\subset \mathbb R^n \rightarrow \mathbb R$. 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 $\mathbb R^n$.

Answer 1

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 $m = \min_{x \in [a,b]^n} f(x)$, $M = \max_{x \in [a,b]^n} f(x)$
• and choose $x_m$ such that $f(x_m) = m$ and $x_M$ such that $f(x_M)=M$

$f(t) := g(tx_m + (1-t)x_M)$ is continuous from $[0,1] \rightarrow [m, M]$