Real analysis: are things really more complicated than they seem

Question

I'm studying real analysis out of Pugh's book. Consider this theorem:

Theorem: Let $f$ be a continuous function defined on $[a,b]$. Then $f$ has an absolute maximum and minimum.

This seems obvious, but it isn't as easy to prove as our intuition suggests (I have Pugh's proof in mind). This is typical in real analysis. Furthermore, the proof that Pugh gives doesn't explain why the theorem is intuitively obvious to us.

Question: Is there a "pathological" continuous function that illustrates why this theorem isn't as easy to prove as our intuition suggests? That is to say, is there an example that can make us doubt our intuition regarding this particular theorem?

Answer 1

It may help to note that a relaxation in any of the conditions of the theorem mean that it's no longer true.

• If the interval is not closed, the theorem doesn't hold. For example, \begin{align*} f \colon (0,1] &\to \mathbb R \\ x &\mapsto \tfrac 1x\end{align*} has no absolute maximum.
• If the interval is not finite, the theorem doesn't hold. For example, \begin{align*} f \colon [1, \infty) &\to \mathbb R \\ x &\mapsto \tfrac 1x \end{align*} has no absolute minimum.
• If the function is not continuous, the theorem doesn't hold. For example, \begin{align*} f \colon [0,1] &\to \mathbb R \\ x &\mapsto \begin{cases} x & x \in (0, 1], \\ 1 & x=0\end{cases}\end{align*} has no absolute minimum.

So perhaps it's not as obvious as it first seems: every part of the theorem is important and cannot be relaxed.