Is this map chain-transitive Let $f:\mathbb R^2→\mathbb R^2$ be a continuous self-map and let $δ$ be a positive real number. A (finite or infinite) sequence $(x_{n})_{n≥0}$ is a $δ$-chain if $$d(f(x_{n}),x_{n+1})<δ$$ for all $n$.
The map $f$ is called chain-transitive if for every $x,y∈\mathbb R^2$ and every $δ>0$ there is a finite $δ$-chain $x₀,...,x_{n}$ such that $x₀=x$ and $x_{n}=y$.
Assuming a map g verify: $g^{k}(x)$ tends to infinity as $k→∞$ for all $x \in \mathbb R^2$, i.e., the successive iterations of $x$ by $g$ goes to infinity, or the orbit of the points $x \in \mathbb R^2$ diverge. 
My question is: is this map $g$ chain-transitive?

Such a $g$ might not be Chain transitive. 
Counter-example: $g(a, b) = (a+1, b)$. Then there does not exists a $\delta$ transitive chain between $x = (0,0)$ and $y = (1,0)$ whenever $\delta <1/2$: No matter what one choose $x_n = (a_n, b_n)$, one has $a_n \ge n/2$.  
