Find all the solutions such that $\frac{∂^2g}{∂u∂v} = 0$

Question

a) Find all the solutions $g : \mathbb R^2 → \mathbb R$ that have partial derivatives up to the second order that are continuous on $\mathbb R^2$ and such that $\frac{∂^2g}{∂u∂v} = 0$

My attempts:

a) since all partial derivatives of $g$ are continuous, then $\frac{∂^2g}{∂u∂v} = \frac{∂^2g}{∂v∂u}$.

If I set $g(u,v) := h(u)$ and I fix $v$, where $(u,v) \in \mathbb R^2$, then $\frac{∂^2g(u,v)}{∂v∂u} = \frac{∂^2h(u)}{∂v∂u} = \frac{∂}{∂v}(\frac{∂h(u)}{∂u}) = 0$

And since the partial derivatives are continuous, then $\frac{∂}{∂v}(\frac{∂h(u)}{∂u}) = \frac{∂}{∂u}(\frac{∂h(u)}{∂v}) = 0$

But I got stuck here. How to continue from there?

Answer 1

$\frac{\partial^2g}{\partial u \partial v} = 0$ Integrate: $\frac{\partial g}{\partial u} = f(u)$ Integrate again: $g(u,v) = h(u)+s(v)$. Where $s$ is any function of $v$ and $h$ is any function of $u$.