How many paths exist between two points in the plane? Fix distinct $a,b \in \mathbb{R}^2$. In terms of cardinality (say, beth numbers), how many distinct continuous functions $f : [0,1] \rightarrow \mathbb{R}^2$ satisfying $f(0)=a, f(1)=b$ are there? (Presumably, the answer is independent of which $a,b$ are chosen.)
It seems clear that the answer is either $\beth_1$ or $\beth_2$.

There are only $\beth_1$ continuous functions from $[0,1]$ to $\Bbb R^2$, so there cannot be more than $\beth_1$, to see this note that every two continuous functions agreeing on a dense set are equal, as $[0,1]$ is separable there is a countable set which "decides" the values of the function. So there are no more than $(\beth_1)^{\aleph_0}=\beth_1$ continuous functions.
Clearly there are $\beth_1$, because take any $c\in\Bbb R^2$, there is a piecewise-linear path such that $f(0)=a,\ f(\frac12)=c,\ f(1)=b$. There are $\beth_1$ such $c$, therefore there are exactly $\beth_1$ many paths.
