If $K$ is compact and $f\colon K\to\Bbb R$ has the intermediate value property, does $f$ attain its extrema?

Question

Assume that $K\subset \Bbb R$ is compact and $f\colon K\to\Bbb R$ is a Darboux function, i.e. it has the intermediate value property. Set $$ m:=\inf_K f, \quad M:=\sup_K f.$$

As discussed here, $(m,M) \subset f(K)$. Now for my questions:

1.) Are $m$ and $M$ finite?

2.) Are there $x,y\in K$ with $f(x)=m$ and $f(y)=M$, provided that these values are finite?

3.) If not, can one give additional assumptions (other than continuity) under which $f$ attains its extrema?

I have never really worked with Darboux functions and I know little to nothing about them. The motivation behind this question is simply curiosity.

Answer 1

Let $f$ be given by the Conway Base 13 Function

$f$ not only satisfies the IVP, but it actually maps every open (and hence every closed) interval to the whole real line, i.e $f([a,b]) = \mathbb{R}$ for any $a,b \in \mathbb{R}$. Thus, $f$ is defined everywhere with the IVP, but is nowhere continuous and does not have any maxima or minima on any closed interval. I can provide proof of these facts, however it will take me a bit.

Answer 2

On $K=[0,1],$ define

$$f(x)=\begin{cases} (1-x)\sin 1/x,&x\in (0,1]\\0,&x=0\end{cases}.$$

Then $f$ has the IVP, but $\inf f =-1,\sup f=1$ are never attained.