On the sectionwise connectedness of a contingent

Abstract

Let X be a real normed space and let f: ℝ → X be a continuous mapping. Let T _f (t ₀) be the contingent of the graph G(f) at a point (t ₀, f(t ₀)) and let S ⁺ ⊂ (0,∞) × X be the “right” unit hemisphere centered at (0, 0 _X ). We show that

• 1. If dimX < ∞ and the dilation D(f, t ₀) of f at t ₀ is finite then T _f (t ₀) ∩ S ⁺ is compact and connected. The result holds for $T_f (t_0 ) \cap \overline {S^ + }$ even with infinite dilation in the case f: [0,∞) → X.

• 2. If dimX = ∞, then, given any compact set F ⊂ S ⁺, there exists a Lipschitz mapping f: ℝ → X such that T _f (t ₀) ∩ S ⁺ = F.

• 3. But if a closed set F ⊂ S ⁺ has cardinality greater than that of the continuum then the relation T _f (t ₀) ∩ S ⁺ = F does not hold for any Lipschitz f: ℝ → X.