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For a regular cardinal κ, a formula of the modal μ-calculus is κ-continuous in a variable x if, on every model, its interpretation as a unary function of x is monotone and preserves unions of κ-directed sets. We define the fragment Cℵ1(x) of the modal μ-calculus and prove that all the formulas in this fragment are ℵ1-continuous. For each formula ϕ(x) of the modal μ-calculus, we construct a formula ψ(x)∈Cℵ1(x) such that ϕ(x) is κ-continuous, for some κ, if and only if ϕ(x) is equivalent to ψ(x). Consequently, we prove that (i) the problem whether a formula is κ-continuous for some κ is decidable, (ii) up to equivalence, there are only two fragments determined by continuity at some regular cardinal: the fragment Cℵ0(x) studied by Fontaine and the fragment Cℵ1(x). We apply our considerations to the problem of characterizing closure ordinals of formulas of the modal μ-calculus. An ordinal α is the closure ordinal of a formula ϕ(x) if its interpretation on every model converges to its least fixed-point in at most α steps and if there is a model where the convergence occurs exactly in α steps. We prove that ω1, the least uncountable ordinal, is such a closure ordinal. Moreover we prove that closure ordinals are closed under ordinal sum. Thus, any formal expression built from 0, 1, ω, ω1 by using the binary operator symbol + gives rise to a closure ordinal.