How can we define the supremum norm on the space of continuous functions if they're not necessarily bounded?

Question

I frequently see the uniform norm described as a norm on the space of continuous functions $C(Y, X)$ where $Y$ is a topological space and $X$ is a metric space, especially in the context of the uniform limit theorem. How is it possible to define such a norm when functions which are continuous need not be bounded?

Specifically, in $C(\mathbb{R}, \mathbb{R})$ the function $f(x) = x$ is continuous and yet $$ \|f\|_{\infty} = \sup_{x \in \mathbb{R}} |f(x)| $$ is not well defined (by the definition of a norm which requires it to be a real valued function on the space).

Answer 1

1. The uniform norm is typically defined on a space of bounded functions.
2. The norm is a function that takes an argument from it's domain and produces a real number out of it. If $f$ is not bounded, there is no way it is a part of a domain of the sup norm.