A Four-Function Functional Equation

Algebra Level 5

\[\large f(x)+g(y)=h(x)k(y)\]

Let the real-valued functions \(f, g, h, k\) be defined on \(\left[0, 1\right],\) such that they satisfy the previous functional equation for all possible values of \(x\) and \(y\) in \(\left[0, 1\right],\) \(f(0)=h(0)=1,\) and \(k\) is not a constant function.

If \(M\) represents the largest possible value of \(f(1)\) and \(m\) the smallest value of \(f(1),\) find \(M-m.\) Enter 666 if either \(M\) or \(m\) does not exist.

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