Define v : [ s$_{1}$, s$_{2}$ ] → R 2 by Note that v ( s ) is in the tangent cone of ∂ Ω at γ ( s ) so that ∂ Ω ∩ D ( γ ( s ) , r ) is graph-like using the orientation given by v ( s ). Define φ ( s ) : [ s$_{1}$, s$_{2}$ ] → R by φ ( s ) = 〈 v ( s ) , p − γ ( s ) 〉 . Note that from γ ( s$_{1}$ ) both v ( s$_{1}$ ) and p − γ ( s$_{1}$ ) are directions pointing into the circle so φ ( s$_{1}$ ) > 0. Similarly, v ( s$_{2}$ ) points out and p − γ ( s$_{2}$ ) points in so that φ ( s$_{2}$ ) < 0. Observe that v (and therefore φ ) is piecewise continuous since γ is piecewise C $^{1}$. By a piecewise continuous analogue of the intermediate value theorem, there exists ¯ s ∈ [ s$_{1}$, s$_{2}$ ] such that By continuity of the inner product and γ , we have Similarly, lim$_{t}$$_{→}$$_{¯}$ s + φ ( t ) = 〈 lim$_{t}$$_{→}$$_{¯}$ s + γ $^{′}$( t ) , p − γ (¯ s ) 〉 If γ is differentiable at ¯ s , then φ (¯ s ) = lim$_{t}$$_{→}$$_{¯}$ s φ ( t ) = 0 and we have our contradic- tion. Otherwise, let w$_{1}$ = lim$_{t}$$_{→}$$_{¯}$ s − γ $^{′}$( t ) and w$_{2}$ = lim$_{t}$$_{→}$$_{¯}$ s + γ $^{′}$( t ). As both w$_{1}$ and w$_{2}$ are in the convex tangent cone of ∂ Ω at γ (¯ s ), any positive linear combination of them is as well. Letting ψ ( λ ) = λw$_{1}$ + (1 − λ ) w$_{2}$ , we have Noting that ψ is continuous in λ , we apply the intermediate value theorem to 〈 ψ (0) , p − γ (¯ s ) 〉 ≤ 0 ≤ 〈 ψ (1) , p − γ (¯ s ) 〉 . 88 lim t → ¯ s − φ ( t ) = 〈 lim t → ¯ s − γ $^{′}$( t ) , p − γ (¯ s ) 〉 . lim t → ¯ s − φ ( t ) ≤ 0 ≤ lim t → ¯ s $_{+}$φ ( t ) . v ( s ) =      lim$_{t}$$_{↓}$$_{s}$$_{1}$ γ $^{′}$( s ) , s = s$_{1}$, lim$_{t}$$_{↑}$$_{s}$ γ $^{′}$( s ) , s ∈ ( s$_{1}$, s$_{2}$ ] .