k-TO-l FUNCTIONS ON ARCS FOR k EVEN 1. eitherf((x,p))çz(f(x),f(p))
... f(x) = f(x') such that no point of (x, p) maps to f(x'). Part 1 is true for this x. Note that the part 1 property implies that each point in (0,1) is either a crossing point, a local maximum, or a local minimum for the graph of /. Now suppose part 2 is false and suppose f(x) < f(p). Then there is an ...
... f(x) = f(x') such that no point of (x, p) maps to f(x'). Part 1 is true for this x. Note that the part 1 property implies that each point in (0,1) is either a crossing point, a local maximum, or a local minimum for the graph of /. Now suppose part 2 is false and suppose f(x) < f(p). Then there is an ...
1. Manifolds with boundary Definition 1.1. An n
... Proof. Let x ∈ ∂X, we need to show that in ∂X there is a neighborhood Vx of x homeomorphic to Rn−1 . Towards this goal, pick a neighborhood (Ux , ϕx ) of x where ϕx : Ux → H n is a / Rn−1 × {0} then there is some ε > 0 such that Bϕx (x) (ε) ⊆ H n . homeomorphism. If ϕx (x) ∈ But then ϕ−1 (Bϕx (x) (ε ...
... Proof. Let x ∈ ∂X, we need to show that in ∂X there is a neighborhood Vx of x homeomorphic to Rn−1 . Towards this goal, pick a neighborhood (Ux , ϕx ) of x where ϕx : Ux → H n is a / Rn−1 × {0} then there is some ε > 0 such that Bϕx (x) (ε) ⊆ H n . homeomorphism. If ϕx (x) ∈ But then ϕ−1 (Bϕx (x) (ε ...
