Real function continuous on closed interval implies it is bounded - over-simple proof??

Real function continuous on closed interval implies it is bounded -
over-simple proof??

So here is my proof, which after looking up others seems to be too simple,
or not rigourous enough, though I don't see why (hence I am asking!):
We take the contrapositive, and so prove that if $f$ is unbounded on
$[a,b]$, then it is not continuous on this interval.
By hypothesis, $\exists c \in [a,b] : \displaystyle\lim_{x\to c} f(x) =
\infty \implies f(c)$ is undefined. (or, $ \displaystyle\lim_{x\to c} f(x)
\not= f(c)$ so we get discontinuity instantly)
Since $f(c)$ doesn't exist, the definition of continuity cannot be applied
so $f$ must be discontinuous at at least $x=c$, as required.