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Let there exist a mystery function ƒ. ƒ accepts exactly 2 arguments, A & B.

As B approaches A, ƒ approaches A, at a simple exponential growth rate E. As B approaches 0, ƒ approaches the mean of A & B, at a simple exponential decay rate D. E is steeper than D (i.e., D is damper than E). Doesn’t matter how much.

Just for practically’s sake… A is always greater than B, and A & B aren’t necessarily codependent (i.e., they may or may not be independent).

QUESTION: Find any continuous function 𝐶 that approximates ƒ for As and Bs between 0 and 1.

The best notation I can muster…

  •   ∃! ƒ: ƒ(α, β) = Q
  •   ∃  𝐶: 𝐶(α, β) ≈ Q
  •   𝑙𝑖𝑚(β→α): ƒ(α, β) = α at rate E
  •   𝑙𝑖𝑚(β→0): ƒ(α, β) = ½(α+β) at rate D
  •   𝑙𝑖𝑚(p→q): 𝐶(p) = 𝐶(q)
  •   E = a(1+r)^x
  •   D = k(1-s)^z
  •   r ≠ s
  •   |E’(n)| < |D’(n)|
  •   ¬ (α ⇔ β)

Find any 𝐶 | 𝐶(α, β) ≈ ƒ(α, β) for 0<β≤α<1.

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The first half of your article reads as giberish to me. If you can't handle your english, please don't use all this proof terminology. –  Ethan Jan 30 '13 at 1:53
    
What do you mean "approaches $A$ at a simple exponential growth rate"? Can't you just splice all the data given together (i.e., for fixed A it is of the form $g_A(B) = ...$, for fixed $B$ it is of the form $h_B(A) = ...$, perhaps it is of the form $g_A(B) \cdot h_B(A)$ and work from there)? –  vonbrand Jan 30 '13 at 2:12
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