TITLE: Path Connectedness and fixed points
QUESTION [5 upvotes]: We have the following given to us,
Let $α, β \colon [0, 1] \to [0, 1]$ be (not necessarily continuous) functions such that

$α(x) ≤ β(x)$, for all $x ∈ [0, 1]$.
The set $K = \{\,(x, y); α(x) ≤ y ≤ β(x)\,\}$ is closed in $\mathbb R^2$
.

Show that there exists $t ∈ [0, 1]$ such that $(t, t) ∈ K$.
We think that if $K$ is closed, then it must be path-connected. In which case, we have a continuous function from $[0,1]$ to $[0,1]$ that is contained in $K$. Therefore can use the intermediate value theorem to show that this function must have a fixed point, which would conclude our argument.
Our difficulty is that we are unable to prove that $K$ is actually path-connected, even though every example of $α$ and $β$ that we can think of satisfying the required properties make $K$ path connected.

REPLY [2 votes]: You only need that $K$ is connected, so suppose $K$ can be divided into $2$ nonempty disjoint closed sets $A,B$.
$K$ is composed of vertical segments and since they are connected, every such segment is entirely in $A$ or in $B$. Therefore the projection of $A$ and $B$ onto the $x$ axis divides the interval $[0,1]$ into $2$ nonempty disjoint closed sets, which is not possible.