Continuity and Compactness, Is this mapping onto? (Rudin)

Question

Here the point $p=(1,0) \in Y$ but there is no $t \in X$ s.t. $f(t)=p$, is it still correct to say that $f$ is mapping of $X$ onto $Y$?

Also, can we say the reason why $f^{-1}$ fails to be continuous is that simply $X$ has no such $t$ so that the point $p$ is not mapped to anything?

Answer 1

You are wrong: $f(0)=(1,0)$. And $f^{-1}$ is discontinuous because it is discontinuous at $(1,0)$.