In which cases, there is no continuous map from A onto B?

Question

(a) $A=[0,1]\cup[2,3], B=\{1,2\}$

(b) $A=(0,1), B=[0,1]$

(c) $A=\mathbb{Q}, B=\mathbb{Q}$

(d) $A=(0,1)\cup(2,3), B=\{1,3\}$

It was clear for (b) as it was already asked numerous times on this site.

For (c), I took identity map.

For (d), We can send $(0,1)$ to $1$ and $(2,3)$ to $3$. Map is clearly onto and into a discrete space. It is continuous as inverse image of each singelton is open.

What about (a)?

Answer 1

a) $x\mapsto \begin{cases}1&x<\sqrt2\\2&x>\sqrt 2\end{cases}$

b) $x\mapsto \frac{1+\sin 42x}2$

c) $x\mapsto x$

d) $x\mapsto \lceil x\rceil$

Answer 2

a) goes the same as (d): map each interval to a separate point.

Note that $[0,1]$ is closed and open in $[0,1] \cup [2,3]$.