A Four-Function Functional Equation

Algebra Level 5

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

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

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

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