boundedness and continuity on a compact set
A set of complex numbers is bounded if it fits inside some disk of finite radius — there is a number M with |z| <= M for every z in the set. A function f is bounded on a set if its output set is bounded, that is |f(z)| <= M throughout. Boundedness is a modest but constantly used notion: it says the function never runs off to infinity over that region.
The reason this pairs with continuity is one of the workhorse theorems of analysis. A set in the plane is compact (in the sense that matters here) exactly when it is both closed (it contains its boundary) and bounded — that is the Heine-Borel description. The theorem then states: a continuous function on a compact set is automatically bounded, and moreover its modulus |f| actually attains a maximum and a minimum value on the set. Nothing escapes to infinity, and the extremes are genuinely reached, not merely approached.
This compactness-gives-boundedness principle is the quiet engine behind several headline results. The maximum modulus principle says that on a compact region a holomorphic f attains its largest |f| on the boundary; Liouville's theorem (a bounded entire function is constant) is a boundedness statement; and many existence proofs work by extracting a convergent subsequence from a bounded family, which is compactness in action. Whenever you see 'on a closed bounded region', expect a compactness argument nearby.
On the closed unit disk |z| <= 1, the continuous function f(z) = z^2 + 1 is bounded: since |z| <= 1, we get |f(z)| <= |z|^2 + 1 <= 2, and the maximum |f| = 2 is actually attained, for instance at z = 1.
On a closed and bounded set, a continuous function is bounded and its modulus attains its extremes.
Both halves of 'closed and bounded' are essential. On the open disk |z| < 1 the continuous function 1/(1 - z) is unbounded as z approaches the boundary, and on the unbounded real line a continuous function like x need not be bounded at all.