# In which of the following cases,there is no continuous function $g$ from the set $A$ onto the set $B\,\,$?

I am stuck on the following problem which I came across in a recent entrance exam:

In which of the following cases,there is no continuous function $$g$$ from the set $$A$$ onto the set $$B$$ ?

1. $$A=[0,1]\,\,,B=\Bbb R$$

2. $$A=(0,1)\,\,,B=\Bbb R$$

3. $$A=(0,1)\,\,,B=(0,1]$$

4. $$A=\Bbb R\,\,,B=(0,1)$$

Can someone explain me how to tackle this?

• I think just 1...$(0,1)$ and $\mathbb R$ are homeomorphic, so 2 and 4 are out. and for 3 just take a function whose graph is say a 'bump' – uncookedfalcon Jun 26 '13 at 3:05

3. Try folding $(0,1)$ in half at $\frac12$ and then stretching it a bit.