In Baby Rudin chapter 7 the author introduced the $\mathscr C(X)$ to denote the class of continuous functions on a metric space $X$, saying that:
If $X$ is a metric space, $\mathscr C(X)$ will denote the set of all complex-valued, continuous, bounded functions with domain $X$.
I'm wondering if the "boundedness" is required in general (although it is required for Rudin's usage here, see edit). I think I've been always seeing and probably even using notations like $1/x\in\mathscr C(0,1)$ or $x^2\in\mathscr C(\Bbb R)$ etc.
So is there a standard usage of the $\mathscr C$ notation? If not, then what is the more commonly used definition of the $\mathscr C$ notation?
Best regards.
EDIT$\quad$ In the specific context where Rudin introduced the notation, the boundedness is required, because he used $\mathscr C(X)$ to denote a linear function space equipped with the supremum norm. But my question is that is it also necessary in the general usage?
If $X$ is any metric space, then the set $\mathscr{C}(X,\mathbf{R})$ of continuous $\mathbf{R}$-valued functions on $X$ makes perfect sense, but unless $X$ is compact, in which case boundedness is not automatic, the supremum norm will not generally be well-defined on this space, because continuous functions need not be bounded in general. If one restricts to the subset $\mathscr{C}^b(X,\mathbf{R})$ of bounded continuous functions, then one has the supremum norm. In general, there are a number of topologies one might put on $\mathscr{C}(X,\mathbf{R})$, but the most commonly-used one, at least if $X$ is locally compact, is probably the compact open topology.
But, again, to answer your question, yes, the space of continuous functions on a metric space is a perfectly reasonable object to consider. But you can't, in general put the supremum norm on it.