Could I define the supremum of a function, given that that function is not continuous?

Question

Starting from a function $f(x)$ which is not continuous on $\mathbb{R}$, it that possible to "define" objects like $\sup\limits_x f(x)$, $x\in\mathbb{R}$?

Answer 1

Yes of course. Let $f(x)=1$ if $x\in\Bbb Q$ and $f(x)=0$ elsewhere. Then $\sup_x f(x)=1$.

To be more precise, $Im(f)=\{f(x):\ x\in\Bbb R\}$ is a subset of $\Bbb R$ assuming that $f:\Bbb R\to\Bbb R$.

If $f$ is a bounded (from above) function (not necessarily continuous), then $Im(f)$ wolud be a bounded set. So there exists $\sup Im(f)\in\Bbb R$.

If $f$ is (upper) unbounded, then $Im(f)$ is unbounded and you can set that $\sup Im(f)=+\infty$.

Answer 2

If you define $\sup_xf(x)$ as $\sup\{f(x)\mid x\in\Bbb R\}$, then this has nothing to do with the continuity of $f$. If, for instance, $f(x)=x$, then $\sup_xf(x)$ doesn't exist. And if$$f(x)=\begin{cases}1&\text{ if }x\in\Bbb Q\\0&\text{ otherwise,}\end{cases}$$then $f$ is discontinuous everywhere, but $\sup_xf(x)=1$.