Question

Assume that f is a continuous function from the real numbers to the real numbers such that for all numbers x,y, and a it is true that f(x+y)=f(x) + f(y) and that f(ax)=af(x). Further assume that f(1) = pi. Find f'(-1). Please explain your reasoning.

I am so confused. Ive read many times but can't figure out where or how to begin. Please help!

Answer 1

if f(ax) = af(x), f(x) = kx for some k.
check:
f(ak) = k(ax) = a(kx) = af(x)
f(x+y) = k(x+y) = kx+ky = f(x)+f(y)

So, since f(1) = pi, f(x) = pi*x
f'(-1) = pi

In fact, f'(x) = pi for any x.