Proof that supremum doesn' exist

Question

We know that if $f:[a,b]\longrightarrow\mathbb{R}$ is continuous, then the function is bounded.

$c \in [a,b]$

$c = \sup\{z \in (a,b]: f\text{ is bounded on }[a,z)\}$

If function is continuous on interval $[a,+\infty)$ how can we prove that supremum doesn't exist.

Answer 1

You can't prove it, since it may exist. If, say, $f$ is constant, that supremum does exist.

But it doesn't always exist. Take $f\colon[0,\infty)\longrightarrow\Bbb R$ defined by $f(x)=x$, for instance.

Answer 2

A supremeum need not exist, as witnessed by the continuous function $x\mapsto x$. A supremum may exist as witnessed by the function $x\mapsto 42$.