Is there a bijective continuous map from $[0,\infty)$ to $\mathbb{R}$?

Question

Whether such a function exists or not, I am trying to prove. The converse is false, that is , there is no continuous bijection from $\mathbb{R}$ to $[0,\infty)$

This is more or less the same question as this https://math.stackexchange.com/questions/8149/points-of-discontinuity-of-a-bijective-function-f-mathbbr-to-0-infty?rq=1 see if it helps — AymaneMaaitat, Apr 23 '20 at 15:49
An injective continuous map on an interval of $\mathbb R$ is either increasing everywhere or decreasing everywhere. — GEdgar, Apr 23 '20 at 15:52

score 2 · Answer 1 · answered Apr 23 '20 at 15:50

2

Hint: $[0,\infty)\setminus\{0\}$ is connected, whereas $\Bbb R\setminus\{f(0)\}$ isn't.

answered Apr 23 '20 at 15:50

José Carlos Santos

427,504

score 2 · Accepted Answer · answered Apr 23 '20 at 16:00

A different route.

Suppose $f : [0,\infty) \rightarrow \mathbb{R}$

Either $\forall x > 0 : f(x) > f(0)$ or $\forall x > 0 : f(x) < f(0)$.

If not then $\exists x, y > 0: f(x) > f(0) \land f(y) < f(0)$. The intermediate value theorem now says that there is a value $z$ between $x$ and $y$ with $f(z) = f(0)$ contradicting bijectivity.

Is there a bijective continuous map from $[0,\infty)$ to $\mathbb{R}$?

2 Answers2