continuous onto map from $(0,1)\to (0,1]$

Question

I need to know whether There exists any continuous onto map from $(0,1)\to (0,1]$

could any one give me any hint?

See also: Continuous function from $(0,1)$ onto $[0,1]$ and other question linked there. — Martin Sleziak, Nov 06 '20 at 11:20

score 13 · Accepted Answer · answered Jul 28 '13 at 12:32

13

Hint: $(0,1) = (0,\frac 12] \cup [\frac 12, 1)$. Can you map each part onto $(0,1]$?

answered Jul 28 '13 at 12:32

martini

No sir, I have no Idea – Myshkin Jul 28 '13 at 12:34
1

hint: $\times 2$ – Dan Rust Jul 28 '13 at 12:36
@TaxiDriver It's a disjoint union of semi-closed intervals. – Git Gud Jul 28 '13 at 12:36
6

Taxi: you have no idea how to send $(0,\frac12]$ continuously to $(0,1]$? Really no idea? – Did Jul 28 '13 at 12:37
Yes $f(x)=2x$????? – Myshkin Jul 28 '13 at 12:38
@GitGud Ya, but nothing coming in my mind now how to use your hint – Myshkin Jul 28 '13 at 12:40
1

@TaxiDriver Yes, you do. – Git Gud Jul 28 '13 at 12:42
Now you've achieved "onto" you only need to maintain "continuous" – Mark Bennet Jul 28 '13 at 12:42
1

@TaxiDriver ... and now: can you map $[\frac 12, 1)$ onto $(0,1]$? – martini Jul 28 '13 at 12:42
1

@Taxi Driver no, you're joking! – W_D Jul 28 '13 at 12:43
1

okay so Here is the Map $f(x)=2x; x\in (0,{1\over 2}]$ and $f(x)=1;x\in [{1\over 2},1)$ – Myshkin Jul 28 '13 at 12:44
Hallelujah! $ $ – Did Jul 30 '13 at 10:06

rurouniwallace · Answer 2 · 2013-07-28T13:26:34.220

6

Find a polynomial that:

$$f(x)=-4(x^2-x),x\in(0,1)$$

edited Jul 28 '13 at 13:26

answered Jul 28 '13 at 12:58

rurouniwallace

score 5 · Answer 3 · answered Jul 28 '13 at 12:46

5

From The Hint of Martini the Map $f(x)=2x; x\in (0,{1\over 2}]$ and $f(x)=1;x\in [{1\over 2},1)$ will work

answered Jul 28 '13 at 12:46

Myshkin

score 4 · Answer 4 · answered Dec 03 '14 at 18:06

4

$f(x)=|\sin (\pi x)|$ will work

answered Dec 03 '14 at 18:06

Mathronaut

1

You can remove the absolute value since the function $\sin(\pi x)$ is positive on the interval $(0,1)$. – Angelo May 23 '21 at 19:47

4 Answers4