# Math Help - Simple Extreme Value Theorem Question

1. ## Simple Extreme Value Theorem Question

I have asked too many hard questions recently now an easy one.

Let $f: [a,b] \rightarrow \mathbb{R}$ be continuous. Then $\exists u,v \in [a,b]$ such that $f(u) \le f(x) \le f(v)$ for all $x \in [a,b]$.

What if we have a function going to a subset of R, e.g. [-6, 7] or something, can we still use the EVT?

2. Yes, it still works. What we have is that $[f(u),\,f(v)]\subseteq[-6,\,7].$ The range of the function is $[f(u),\,f(v)]$ so any codomain $S\subseteq\mathbb R$ such that $[f(u),\,f(v)]\subseteq S$ will be fine.