Find all continuous functions f, g, h : R → R for all real values of x and y
satisfying the functional equation
f(x + y) = g(x)+h(y) .
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1 Answers
66That's the so called Pexider's functional equation.
Setting x=0, we get
f(y) = g(0) + h(y).
Let g(0) = a, so that we get h(y) = f(y) - a, or which is the same as
h(x) = f(x) - a for all x.
Similarly by setting y=0 and h(0) = b we get
g(x) = f(x) -b.
So the equation now becomes
f(x+y) = f(x) + f(y) -a -b
Next we define F(x) = f(x) -a -b for all x. The above equation then becomes
F(x+y)+a+b = F(x) + a + b + F(y) + a+b -a - b
i.e.
F(x+y) = F(x) + F(y)
which is the Cauchy's equation. The continuity of f implies the continuity of F so the general solution is
F(x) = cx for some constant c.
Hence we get the general solution for the original given equation as
f(x) = c x + a + b
g(x) = c x + a
h(x) = c x + b
