Example 6. If properties P1 and P2 are satisfied, then the graph metric (2)
coincides with the intrinsic orbifold metric (3).

Equation (3) states that d (·|·) is the length of a minimizing geodesic of X
and Y and therefore an intrinsic metric, because it coincides with the infimum of the length of all admissible curves from X to Y . In addition, we find that the topology of XΓ induced by the metric d coincides with the quotient topology induced by the topology of the Euclidean space X .

## 3.4 Orbifold Functions

Suppose that Q = (X *, Γ, π*) is an orbifold. An *orbifold function* is a mapping f : XΓ → R.

The *lift* of f is a function

$${\bar{f}}:{\mathcal{X}}\to\mathbb{R}$$

satisfying ˜f = f ◦ π. The lift ˜f is invariant under group actions of Γ, that is ˜f(x) = ˜f (γ(x)) for all γ ∈ Γ.

We say, an orbifold function f : XΓ → R is continuous (locally Lipschitz, differentiable, generalized differentiable) at X ∈ XΓ if its lift ˜f is continuous
(locally Lipschitz, differentiable, generalized differentiable) at some vector representation x ∈ X. The definition is independent of the choice of the vector representation that projects to X (see Section B.1, Prop. 1 - Prop. 4). For a definition of generalized differentiable functions and their basic properties we refer to Section A.

Example 7. Consider the Riemannian orbifold (X *, Γ, π*) of Example 1-5. The function fY : XΓ → R, X 7→ k(*X, Y* )
for some Y ∈ XΓ is an orbifold function with lift

$${\tilde{f}}_{Y}:{\mathcal{X}}\to\mathbb{R},$$

˜fY : X → R, x 7→ max {hx, yi,hx, γ(y)i},
where y ∈ Y . Analytical properties of f such as continuity and differentiability can be investigated using the lift ˜f of f. For example, if ˜f is differentiable at x ∈ X then it is also differentiable at γ(x) according to Prop. 3. Hence, differentiability of the orbifold function f is well-defined at X.

## 3.5 Gradients And Generalized Gradients Of Orbifold Functions

We extend the notion of gradient and generalized gradient to differentiable and generalized differentiable orbifold functions.