$C(\mathbb R^n) \cap L^\infty(\mathbb R^n)$

Question

im reading some basic notes on pde and more than once the term $C(\mathbb R^n) \cap L^\infty(\mathbb R^n)$ comes up

now isnt the $L$ part redundant since $C=C^0$ is the space of all continuous functions with finite maximum norm?

or is it just the space of continuous functions?

but then what sense does the norm make for unbounded functions?

glad for any help!