# Continuously differentiable function

1. Apr 7, 2014

### Lee33

1. The problem statement, all variables and given/known data

Show that if $f$ is a continuously differentiable real valued function on an open interval in $E^2$ and $\partial^2f/\partial x\partial y=0,$ then there are continuously differentiable real-valued functions $f_1,f_2$ on open intervals in $\mathbb{R}$ such that $f(x,y)=f_1(x)+f_2(y).$

How can I prove this?

2. Relevant equations

None

3. The attempt at a solution

Let $(x_0,y_0)\in E^2$ and integrate twice:

$0=\int_{y_0}^y\int_{x_0}^x\partial_x(\partial_yf(x',y'))dx'dy'=\int_{y_0}^y(\partial_yf(x,y')-\partial_yf(x_0,y'))dy'=f(x,y)-f(x,y_0)-f(x_0,y)+f(x_0,y_0).$

2. Apr 7, 2014

### micromass

Staff Emeritus
This looks fine. Was there any problem with this?

3. Apr 7, 2014

### Lee33

Nope, I was just confirming. Thanks for confirming!