(i) Show that a metric space M is connected if and only if every continuous integer-valued function on M is constant.
(ii) Show that $H = \{(x, y) \in R^2 : x > 0\}$ is connected. By considering the function $f(x, y)/x$, or otherwise, show that there are precisely two continuous functions $f : H \to R$ such that $f(x,y)^2 = x^2$ for all $(x,y) \in H$.
(iii) How many continuous functions $g : R^2 \to R$ are there such that $g(x, y)^2 = x^2$ for all $(x, y) \in R^2$?
I can do (i), and the connectedness of H.
My solution attempt for the rest: If $f(x,y)^2 = x^2$, then $f(x,y)/x = 1$ or $f(x,y)/x = -1$, i.e. $f(x,y) = x$ or $f(x,y) = -x$, which are precisely two continuous functions. Same answer for (iii).
This sounds wrong: it does not make use of any of the facts I proved before in the question (not even connnectedness of H!), and (ii) and (iii) seem setup to yield differing answers, not the same one.
Any ideas as to what I am missing? Hints especially appreciated.
Best wishes,
Leon
Hint : The short answer to (iii) is "There are $2\times 2=4$ solutions."