Suppose that the function $f$:$R^{2}R$ has first$-$order partial derivatives and that

$\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{x}}(x,y)=\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{y}}(x,y)=0,$ for all $(x,y)in{R}^2.$

Prove that the function $f$:$R^{2}Ris$ constant, that $is,$ that there $is$ some number $c$

such that

$f(x,y)=c,$ for all $(x,y)in{R}^{2}f$:$R^{2}Rto$ a line parallel $to$ one $of$ the

coordinate axes $is$ constant.