So this should be a very simple question. The Closed Graph Theorem as stated in Royden is Let $T\colon X \rightarrow Y$ be a linear operator between Banach spaces $X$ and $Y$. Then $T$ is continuous iff it is closed. Now, does this presuppose that the domain of $T$ is either closed, or all of $X$ (which implies it is closed?). In particular, can we only say a continuous linear operator is a closed linear operator if we are considering a closed domain?
For example, consider the identity function in $\Bbb R$ from $(-1,1)\to\Bbb R$. We can take $x_n \rightarrow 1$ such that $x_n \in \Bbb R$ for all $n$, clearly $Tx_n \rightarrow 1 \in\Bbb R$ but $1$ is not in $(-1,1)$ so $T$ is not closed. Is this incorrect?
Basically, all the texts I have looked at have not been explicit about whether we are presupposing the domain to be closed. Also, I keep finding statements like all closed functions are continuous, i.e. that closed operators include continuous operators, but I feel like the above example excludes this. Can somebody verify if I am right? And if I am not, can you explain what is wrong? Thank you!