Suppose that f(x) satisfies the functional equation
f(x + y) = f(x) + f(y)
for all x,y in R (the real numbers). Prove that if f(x) is continuous that f(x) = cx where c is a constant. What can you say about f(x) if it is allowed to be discontinuous?© BrainMass Inc. brainmass.com March 4, 2021, 5:44 pm ad1c9bdddf
f(0) = f(0) + f(0)
Since f is continuous
lim h->0 f(x+h) = lim h->0 f(x-h)
because of the functional equation,this means
lim h->0 f(h) = lim h->0 limf(-h)=0
Also f(2x -x) = f(2x) + f(-x)
means f(2x) = f(x) - f(-x)
and f(2x) = f(x) + ...
The notion of discontinuity in relation to a given function is investigated.