Let $f, g : \mathbb{R}^2 \longrightarrow \mathbb{R}$ 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 statement(s) from below: (A) $f$ is uniformly continuous; (B) $f$ is bounded; (C) The function $(f,g) : \mathbb{R}^2 \longrightarrow \mathbb{R}^2$ is differentiable; (D) If $\partial f/\partial x = \partial g/\partial y$ and $\partial f/\partial y = -\partial g/\partial x$, then the function $\mathbb{C} \longrightarrow \mathbb{C}$ sending $(x + \imath y) \longrightarrow f(x,y) + \imath g(x,y)$ (with $x, y \in \mathbb{R}$) is constant.
Let $f, g : \mathbb{R}^2 \longrightarrow \mathbb{R}$ 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 statement(s) from below:\\
(A) $f$ is uniformly continuous;\\
(B) $f$ is bounded;\\
(C) The function $(f,g) : \mathbb{R}^2 \longrightarrow \mathbb{R}^2$ is differentiable;\\
(D) If $\partial f/\partial x = \partial g/\partial y$ and $\partial f/\partial y = -\partial g/\partial x$, then the function $\mathbb{C} \longrightarrow \mathbb{C}$ sending $(x + \imath y) \longrightarrow f(x,y) + \imath g(x,y)$ (with $x, y \in \mathbb{R}$) is constant.