Let the $f(x,y)$ be a symmetric function where $x\in[0,1], y\in[0,1]$ and $f(x,y)\in[0,1].$
The question is whether the following statement is valid.
\begin{equation} \int_0^1\frac{\partial f(x,y)}{\partial x}dy >0 \Leftrightarrow \frac{\partial f(x,y)}{\partial x} >0 \end{equation}
If its not, could anyone gives me a example.
You did not specify whether the inequality is supposed to hold for all values of $x$. But, assuming you meant that, try $$f(x,y) = \frac34(x+y)-xy.$$