Continuous surjective map Does there exist a continuous surjective map from $\{(x, y)\in \mathbb{R^2} | x^2-y^2=1\}$ to $\mathbb{R}$?
Any help would be greatly appreciated. I was trying to work with taking one branch of the hyperbola and restricting the function to it. But I cannot understand how to proceed much further.

Define $f: \mathbb R^2 \setminus (\{0\}\times \mathbb R)$ by
$$f(x,y) = \begin{cases} x&& x>0\\ x+2 && x<0 \end{cases}$$
Then $f$ is continuous on $\mathbb R^2 \setminus (\{0\}\times \mathbb R).$ Since the hyperbola, let's call it $H,$ is contained in the domain of $f,$ $f$ is continuous on $H.$ Check that $f$ sends the right branch of $H$ onto $[1,\infty),$ and sends the left branch of $H$ onto $(-\infty,1].$ Hence $f|_H$ is a continuous map of $H$ onto $\mathbb R.$
