Suppose there is a function $f:\mathbb{R}_+^n\mapsto \mathbb{R}$. Are there any systematic ways to determine whether the range of $f$ is bounded or not?
For example, there is a function $f(x,y)=-x^2+2y\log(1+x)-y\log(y),x>0,y>0$. How can we prove that it is bounded above (as indicated in the simulation results)?
There isn't any general method. What you usually need to do in practice is like Robert illustrated. The function is bounded on compact sets on which the function is continuous, so you need to focus on the neighbourhoods of points of non-continuity and on trajectories going to infinity. Note that it is easy to make a continuous function $f(x,y)$ such that $f(x,y)\to 0$ as $x,y\to\infty$ along any straight line, and yet $f(x,y)$ is not itself bounded. So it isn't enough to consider straight trajectories. Other methods that might work on odd occasions include writing the function as an integral (which might be obviously bounded when it isn't obvious in the original formulation), and interpreting the function as something already known to be bounded (such as a probability).
