I am trying to find a function $f(x,y)$ with $f:\mathbb{R}^{2} \rightarrow [a, b]$ where $[a, b]$ is equal to $[-1, 1]$, $[0, 1]$, or some other small interval (open intervals are fine as well).
The partial derivative $\displaystyle\frac{\partial}{\partial x} f(x,y)$ should be $x$ and the partial derivative $\displaystyle\frac{\partial}{\partial y} f(x,y)$ should be $y$.
The function $f(x,y)=\frac{1}{2}x^2+\frac{1}{2}y^2$ would have the partial derivatives I want but $\lim_{x\rightarrow\infty} f(x,y) = \infty$ (and the same for $y$) so it does not have the bounds I want.
Is it possible to find such function? If it helps it is possible to restrict $x>0$ and $y>0$.
$f'_x(x,y)=x\Rightarrow f(x,y)=\frac{x^2}{2}+g(y)\Rightarrow y=f'_y(x,y)=g'(y)\Rightarrow g(y)=\frac{y^2}{2}+C\Rightarrow f(x,y)=\frac{x^2+y^2}{2}+C$, where $C$ is a real constant. There are no other options if you want $f$ to be differentiable and at the same time fulfill those conditions for the partial derivatives.
So if you want the function to only map on a specific limited interval, you will have to restrict the domain of the function. Maybe you can restrict to certain limited intervals for $x$ and $y$?
If it would work, you could maybe have a function $f(x,y)=k\cdot(x^2+y^2)+C$ instead of the one you want, choosing suitable intervals for $x$ and $y$ and then adjusting the constants $k$ and $C$?