Consider a function $f\left( \phi \right) :\mathbb{R}^{N}\rightarrow \mathbb{R}
$ that is translation invariant,
\begin{equation*}
f\left( \phi +b\right) =f\left( \phi \right) +b
\end{equation*}
for scalar $b$, which implies
\begin{equation*}
\nabla f\left( \phi \right) \cdot \mathbf{1}=1
\end{equation*}
where $\cdot $ denotes the dot product and $\mathbf{1}$ is a vector of 1's.
Assume the gradient of $f$ always exists. Furthermore, all elements of the
gradient are non-negative (and hence bounded from above by 1).
Does the limit \begin{equation*} \lim_{t\rightarrow \infty }\frac{f\left( \phi t\right) }{t} \end{equation*} exist in general?
Are there necessary and sufficient conditions for its existence?