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Twice Differentiable Pointwise Factor Zero

2425MasterCalculusFunctions

75th Romanian Mathematical Olympiad

Find all pairs of twice differentiable functions $f, g : \mathbb{R} \to \mathbb{R}$, such that $f''$ and $g''$ are continuous, such that $$ (f(x) - g(y)) \cdot (f'(x) - g'(y)) \cdot (f''(x) - g''(y)) = 0, $$ for all $x, y \in \mathbb{R}$.
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