Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We assume that the functions $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ are bounded on $\mathbb { R } ^ { 2 }$. Show that the functions $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ are constant.
Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations:
$$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$
We assume that the functions $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ are bounded on $\mathbb { R } ^ { 2 }$.
Show that the functions $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ are constant.