8.9.12 f(x) – f(y) = (x – y)f' () This functional equation can be rewritten to the form f(2) – {(1) = r (**"). x + y (8.451) %3D x - y 424 Difference Equations Note that if we set x = y + h, then ( flu + h) – f(w) – r (y+;). f' (y + 2 (8.452) 0, we have the identity f'(y) = f'(y). and upon taking Lim h Now, taking the derivatives of x + y ( (a) – f(1y) = (x – 9)s" ("). (8.453) respectively, with x and then y, gives (ro)-r(뿌) -» (3)r(부) x + y x + y '() = ' ()+ (* - u) () " (*) (8.454) %3D and (-)()-)-() • (9) --wr(*) x + y fl" x + y x + y (x – y) f" (*+y (8.455)

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