Let $f,g:\mathbb{R}^2 \to \mathbb{R}$ be two differentiable functions such that $f(x+1,y)=f(x,y+1)=f(x,y)$ and $g(x+1,y)=g(x,y+1)=g(x,y)$ for all $(x,y)\in \mathbb{R}^2$ Choose the correct statements below :
a)$f$ is uniformly continuous
b)$f$ is bounded
c)The function $(f,g):\mathbb{R}^2\to \mathbb{R}^2$ is differentiable
d)If $\frac{\partial f}{\partial x}=\frac{\partial g}{\partial y}$ and $\frac{\partial f}{\partial y}=-\frac{\partial g}{\partial x}$ then the function $\mathbb{C}\to \mathbb{C}$ sending $(x+iy)\to f(x,y)+ig(x,y)$ (with $x,y\in \mathbb{R}$) is constant
if i consider $\phi (x)=f(x,y)$ then $\phi(x+1)=\phi(x)$ now as $f$ is continuous then $\phi$ is also . whhich implies that $\phi$ is constant ...but how prove it?? This is an exam question .Answer is all .
$f$ is uniformly continuous, since it is periodic and is bounded, ( Remark $f(\mathbb{R}^2)=f([0,1]\times [0,1])$ and a continuous functioon defined on a compact space is bounded). For $d$, it is a consequence of Liouville theorem, since the function is bounded.