A continuous linear function must have the form f(x) = ax. Discontinuous linear functions look dreadful.

Assuming that the function f is also continuous I plan to show that f(x) = ax for some real a. Please note that if indeed f(x) = ax then a = f(1) which provides a starting point for the proof. But first let me note that (*) contains an unknown which, as we are going to establish, is equal to f(x) = ax. In other words, (*) serves as an example of a functional equation - an equation whose unknown is a function.

Proof

The proof proceeds in several steps.

1. x is 0.

   f(0) = f(0 + 0) = f(0) + f(0) = 2f(0).

   Therefore f(0) = 2f(0) and finally f(0) = 0.

2. x is negative.

   Let x be negative, e.g., let x + y = 0, where y is positive; so that -x = y. Then

   0 = f(0) = f(x + y) = f(x) + f(y).

   Therefore f(-x) = f(y) = -f(x).

3. x is an integer.

   We have f(2) = f(1 + 1) = f(1) + f(1) = 2f(1). By induction, assume f(k - 1) = (k - 1)f(1). Then

   f(k) = f(1 + (k-1)) = f(1) + (k-1)f(1) = kf(1).

   Let's denote a = f(1). We have shown that for all integers n, f(n) = an.

4. x is rational

   First of all, for any integer n0, we have 1 = n/n. Then, as before, a = f(1) = f(n/n) = nf(1/n). Hence, f(1/n) = a/n = a(1/n). For p = m/n we similarly have

   f(p) = f(m/n) = mf(1/n) = m·a/n = a(m/n) = ap.

   
   
5. x is irrational

   Any irrational number r can be approximated by a sequence of rational numbers p_i. The closer p_i is to r, the closer ap_i is to ar. However, since ap_i = f(p_i) and assuming f continuous we must necessarily get  f(r) = ar.

Continuity of the function is quite essential as it's possible to show [Ref. 1, 2] that the graph of any discontinuous solution to (*) is dense in the plane R². For the sake of reference, the graph of a function f: RR is defined as a set of pairs (x, y), i.e. elements of R² such that y = f(x). Formally, graph(f) = {(x, y)R²: y = f(x)}.

Remark

Generally speaking, a function that satisfies (*) is called additive. The function that satisfies f(x) = ax for some a is said to be homogeneous. A function is said to be linear if it's both additive and homogeneous. We have just shown that a continuous additive function is necessarily linear.

The graph of a linear function is a straight line whose (linear) equation may be obtained in different forms depending on the manner in which the line is defined.

References

1. B. R. Gelbaum and J. M. H. Olmsted, Counterexamples in Analysis, Holden-Day, 1964
2. B. R. Gelbaum and J. M. H. Olmsted, Theorems and Counterexamples in Mathematics, Springer-Verlag, 1990