In axiomatic set theory, a function f : Ord → Ord is called normal (or a normal function) if it is continuous (with respect to the order topology) and strictly monotonically increasing.
But f (α) = α + 1 is not normal because it is not continuous at any limit ordinal; that is, the inverse image of the one-point open set {λ + 1} is the set {λ}, which is not open when λ is a limit ordinal.
More important examples of normal functions are given by the aleph numbers
Furthermore, for any non-empty set S of ordinals, we have Proof: "≥" follows from the monotonicity of f and the definition of the supremum.
For "≤", set δ = sup S and consider three cases: Every normal function f has arbitrarily large fixed points; see the fixed-point lemma for normal functions for a proof.