content_image/298310.jpg content_image/298309.jpg content_image/298308.jpg content_image/298307.jpg content_image/298306.jpg
( a, b ] if a < b and ̂ ( a, b ] = ( b, a ] if a > b. 

Theorem 5.1 The function φ ( t ) is unpredictable. 

Proof. First of all, we will show that ρ ( φ$_{i}$$_{n}$ , φ ) → 0 as n → ∞ , where ρ is the metric defined by equation (2.1). Let us fix an arbitrary natural number n. The functions φ ( i$_{n}$ + s ) and φ ( s ) satisfy the equation 

<img src='content_image/298310.jpg'>

It is worth noting that for each r ∈ $^{Z}$both of the points ζ$_{r}$ and ζ$_{i}$$_{n}$$_{+}$$_{r}$ − i$_{n}$ belong to the interval ( r, r +1) . ∞ 

Moreover, ‖ ν ( i$_{n}$ + s ) − ν ( s ) ‖ = ‖ d$_{1}$ − d$_{2}$ ‖ for s ∈ ⋃ r = −∞ ̂ ( ζ$_{r}$, ζ$_{i}$$_{n}$$_{+}$$_{r}$ − i$_{n}$ ] , and ‖ ν ( i$_{n}$ + s ) − ν ( s ) ‖ = 0 , otherwise. 

Since for each r = − 2 k$_{n}$, − 2 k$_{n}$ + 1 , . . . , k$_{n}$ − 1 the distance between the points ζ$_{r}$ and ζ$_{i}$$_{n}$$_{+}$$_{r}$ − i$_{n}$ are at most δ$_{n}$, one can verify for each s ∈ [ − k$_{n}$, k$_{n}$ ] that 

<img src='content_image/298309.jpg'>

Hence, we have 

<img src='content_image/298308.jpg'>

The last inequality implies that ρ ( φ$_{i}$$_{n}$ , φ ) → 0 as n → ∞ , i.e., φ ( t ) is a Poisson function. 

Now, let us show the existence of a positive number ϵ$_{0}$ satisfying ϵ$_{0}$ → 0 as ϵ$_{0}$ → 0 such that ρ ( φ$_{i}$$_{n}$$_{+}$$_{j}$$_{n}$, φ$_{j}$$_{n}$ ) ≥ ϵ$_{0}$ for each n ∈ $^{N}$. For a fixed natural number n, using the equations 

and 

<img src='content_image/298307.jpg'>

<img src='content_image/298306.jpg'>

