This is the question:
Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function. Prove that the vector field $F : \mathbb{R}^2 \to \mathbb{R}^2$ defined by $F(x, y) := (f(x + y), f(x + y) )$ is conservative.
My attempt was this: A field is conservative if $\oint_C F\cdot dl= 0$ for any simple closed curve. I took C an arbitrary simple closed curve, so $F$ follows the green's Theorem, then the line integral is $$\oint_{\partial D}F\cdot dl=\iint_D\frac{\partial f(x+y)}{y}-\frac{\partial f(x+y)}{x}$$
and I only get up here, my professor suggest that use the Schwarz's theorem, but I'm not pretty sure if i can use it, because I know that F is continuous because is continuous component by component, but I'm not sure that the derivative is also continuous or either exists, so I'm stuck with that
i appreciate any suggestions or some help with this exercise
Your approach dosen't work as you don't know if $f$ is even differentiable, so Green's theorem cannot apply.
One approach is to define the antiderivative $$ \varphi(x,y) = \int_0^{x+y} f(t) \ dt \, , $$ which is well defined as $f$ is continuous. Then according to the Fundamental Thm. of Calculus (or the Leibniz integral rule), $\partial \varphi/\partial x = f(x+y) = \partial \varphi/\partial y$. It follows that your vector field $F = \nabla \varphi$, and hence is conservative.