Continuous maps from $\Bbb R^2 \to \Bbb R$ Which of the following statements is true ?

*

*There are at most countably many continuous maps from $\Bbb R^2 \to \Bbb R$


*There are at most finitely many continuous surjective maps from $\Bbb R^2 \to \Bbb R$


*There are infinitely many continuous injective maps from $\Bbb R^2 \to \Bbb R$


*There are no continuous bijective maps from $\Bbb R^2 \to \Bbb R$
My Attempt:
If I take a function from $\Bbb R^2 \to \Bbb R$ that maps Circle to Line such that $f(x,y) = \alpha x$ which is continuous where $\alpha \neq 0$ and $\alpha \in \Bbb R$ but cardinality of $\Bbb R$ is $C$ which is uncountable.  So first two options are discarded. Am I right? If not, correct me with good explanation. Also cardinality of $\Bbb R^2 = C^2$ > $C$ = cardinality of $\Bbb R$ . So there does not exist injective map from $\Bbb R^2$ $\to$ $\Bbb R$ Hence option 3 is not true and 4 is only option that is true. Please help me. Thanks in advance
 A: 1, 2. I think you're on the right track here, but what is this circle to line business?
3, 4: A continuous injective map from $\mathbb R^2$ to $\mathbb R$ would take each vertical line in $\mathbb R^2$ to an interval in $\mathbb R$ of positive length. How many pairwise disjoint intervals can there be in $\mathbb R?$
A: For option 1, Take $f(x,y) = c $ $(c \in \Bbb R)$, is a constant function which is continuous from $\Bbb R^2 \to \Bbb R$.But $\Bbb R$ is uncountable. So there exist uncountable many constant continuous functions from $\Bbb R^2 \to \Bbb R$. Hence option 1 is not true.
For option 2, Take $f(x , y) = \alpha x$ $(\alpha \in \Bbb R)$, which is a polynomial and polynomial is continuous function from $\Bbb R^2 \to \Bbb R$. For all $x \in \Bbb R$ there exist $(x , y)$ in $\Bbb R^2$ such that $f(x , y) = \alpha x$. So $f$ is onto. But choice of such $\alpha$ is uncountable. Hence option 2 is not true.
For option 3, Assume that $f$ is continuous and injective. We know that image of connected set is connected set under continuous map. So $f(\Bbb R^2) = Interval$. If we remove some points from the plane $\Bbb R^2 $ then it is still path connected $\implies$ connected but when we remove point from $\Bbb R$ then it is disconnected. Hence option 3 is not true.
For option 4, Bijective map means a map that is injective as well as surjective. But from 3, it is not injective. So it is not bijective . Hence option 4 is true.
