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

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

could any one give me any hint?

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

• hint: $\times 2$ – Dan Rust Jul 28 '13 at 12:36
• Taxi: you have no idea how to send $(0,\frac12]$ continuously to $(0,1]$? Really no idea? – Did Jul 28 '13 at 12:37
• @TaxiDriver Yes, you do. – Git Gud Jul 28 '13 at 12:42
• @TaxiDriver ... and now: can you map $[\frac 12, 1)$ onto $(0,1]$? – martini Jul 28 '13 at 12:42
• @Taxi Driver no, you're joking! – W_D Jul 28 '13 at 12:43

Find a polynomial that:

1. Crosses the x-axis at $x=0$ and $x=1$.
2. Has an absolute maximum of $f(x)=1$.

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

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

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