Doubt about discontinuity

Question

I found this exercise:>Let $f$ be a monotone function, then it can have only a countable number of discontinuities.

I've seen around here a lot of posts about the same but I never get to any conclusion. What happens with Identity function restricted to the set of irrationals? It is monotone, but, isn't it discontinuous at every point of its domain? Which, by the way is uncountable.

I know that if $f$ were defined on an interval, it would be easy to prove, but this is not the case...

Could anyone clarify me please?

Answer 1

Actually, the identity function, restricted to the irrationals, is continuous at every point of its domain. If you want a $\varepsilon-\delta$ proof, then just pick, for any $\varepsilon>0$, $\delta=\varepsilon$. So, it is not a counter-example.

Answer 2

The identity function restricted to the set of irrationals, $f:\mathbb{Q}^c\to \mathbb{R}$ defined by $x\mapsto x$, is continuous on $\mathbb{Q}^c$ and discontinuous on $\mathbb{Q}$, which is countable. I don't see how this is a counter-example to the statement.

To prove this, let $a\in \mathbb{Q}^c$. For any $\varepsilon >0$, choose $\delta =\varepsilon >0$. For any $x \in \mathbb{Q}^c$ such that $|x-a|<\delta $, we have $$|f(x)-f(a)|=|x-a|<\delta =\varepsilon .$$ Thus, we have that $f$ is continuous at the irrationals, i.e. on its domain. Also see that $f$ has removable discontinuity at the rationals since it is not defined at those points. Note that $\mathbb{Q}$ is countable.

Answer 3

I think it may help your curiosity to prove that any monotonic function from any order topologies to another order topology can have only countably many discontinuities. But as you suggest this isn't always true. Consider the long line $\omega_1\times[0,1)$ with the lexigraphical ordering. Define a monotonic map from it to itself as $f(x,y)=(x,\frac{y}{2})$ if $y<\frac{1}{2}$ and $f(x,y)=(x,\frac{y+1}{2})$ otherwise. In this case you can check $f(x,y)$ is monotonic, but has uncountable discontinuities. Now try and find conditions which guarantee the statement to be true, maybe something to do with paracompact, second countable, Lindelof, or seperability.