= Solution
The <chain rule> for <Fréchet derivatives> states that if $f$ is <differentiable> at $a$ and $g$ is <differentiable> at $f(a)$, then
$$
\boxed{D(g\circ f)_a=Dg_{f(a)}\circ Df_a}.
$$
The composition on the right is a composition of <linear maps>, equivalently a product of <Jacobian matrices> in coordinates.
Apply the <chain rule> to the <affine function> $x\mapsto(x,c-x)$. The resulting <derivative> is
$$
\boxed{g_c'(x)=f_x(x,c-x)-f_y(x,c-x)}.
$$
If the two <partial derivatives> agree everywhere, $g_c'=0$. The <mean value theorem> makes $g_c$ constant on $\mathbb R$. Define $h(c)=f(c,0)$, which is itself <differentiable> by the <chain rule>. Taking $c=x+y$ gives $g_c(x)=g_c(c)=h(c)$, and therefore
$$
\boxed{f(x,y)=h(x+y)}.
$$
No <continuity> of the <partial derivatives> beyond the assumed <differentiability> is needed.
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