0

We know that any monotone function from $\mathbb{R} \to \mathbb{R}$ has only countable discontinuities. What about monotone functions on, say $[a,b]^2 \to [a,b]^2$ (i.e., component-wise monotone)?

  1. Do they also have only countable discontinuities ?

  2. Does the result hold for mixed monotonic functions ?

  3. In general, what happens with the number of discontinuities of monotone functions over ordered compact spaces ?

    Thanks in advance for the answers.

Community
  • 1
Rahul
  • 1

1 Answers1

3

The function $[0,1]^2\to[0,1]^2$ $$f(x,y)=\begin{cases}(1,1)&\text{if }x+y>1,\\(0,0)&\text{if }x+y\le 1\end{cases}$$ is componentwise monotone increasing (you could even easily make that strictly increasing) and fails to be continuous (even componentwise) in uncountably many points.

Hagen von Eitzen
  • 384,686