Let a be a real. Thus,

\(\displaystyle{f{{\left({x},{y}\right)}}}={\int_{{{a}}}^{{{x}}}}{g{{\left({t}\right)}}}{\left.{d}{t}\right.}-{\int_{{{a}}}^{{{y}}}}{g{{\left({t}\right)}}}{\left.{d}{t}\right.}={G}{\left({x}\right)}-{G}{\left({y}\right)}\)

\(G(X)=\int_{a}^{X} g(t)dt\)

\(\displaystyle{G}'{\left({X}\right)}={g{{\left({X}\right)}}}\)

Since g is continuous at R,

\(\displaystyle{{f}_{{{x}}}{\left({x},{y}\right)}}={G}'{\left({x}\right)}={g{{\left({x}\right)}}}\)

\(\displaystyle{{f}_{{{y}}}{\left({x},{y}\right)}}={G}'{\left({y}\right)}=-{g{{\left({y}\right)}}}\)