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===Finitely many variables===
In this section it is more convenient to think of φ<sub>α</sub>(β) as a function φ(α,β) of two variables. Veblen showed how to generalize the definition to produce a function φ(α<sub>''n''</sub>,α<sub>''n''−1</sub>,…,α<sub>0</sub>) of several variables, namely:
* φ(α)=ω<sup>α</sup> for a single variable,
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* φ(0,α<sub>''n''−1</sub>,…,α<sub>0</sub>)=φ(α<sub>''n''−1</sub>,…,α<sub>0</sub>), and
* for α>0, γ↦φ(α<sub>''n''</sub>,…,α<sub>''i''+1</sub>,α,0,…,0,γ)
For example, φ(1,0,γ) is the γ-th fixed point of the functions ξ↦φ(ξ,0), namely Γ<sub>γ</sub>; then φ(1,1,γ) enumerates the fixed points of that function, i.e., of the ξ↦Γ<sub>ξ</sub> function; and φ(2,0,γ) enumerates the fixed points of all the ξ↦φ(1,ξ,0). Each instance of the generalized Veblen functions is continuous in the ''last nonzero'' variable (i.e., if one variable is made to vary and all later variables are kept constantly equal to zero).
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