Let $f : \mathbb{R}^2 \to \mathbb{R}$ be a function of two variables. Now, consider the function $g(x) = \int f(x, y)\; dy$. It is known that $g$ has exponential growth, that is, for large values of $|x|$, it behaves like a function of the type $ae^{b|x|}$, where $a, b>0$. I am wondering whether we can say that there exists $y_0$ such that $f(x, y_0)$ also grows exponentially in $x$, or can $f(x, y)$ have subexponential growth for all $y$? To make it very simple: can a family of slowly growing functions be "averaged" to find a function of faster growth?
I suspect that this might not be strictly true, but unable to come up with a counterexample. Any help would be appreciated.
