differentiationproblemssupposeafunctionfisdefinedontherealnumbersthatforeachxandy,f(x+y)=f(x)*f(y)showthata)f(0)=0orf(0)=1b)iffisdifferentiableatanyxandf'(x)=f(0)*f(x)actually,thesecondquestionhastwoparts,the1stisif

Question

differentiation problems
suppose a function f is defined on the real numbers that for each x and y,f(x+y)=f(x)*f(y)
show that
a)f(0)=0 or f(0)=1
b)if f is differentiable at any x and f'(x)=f(0)*f(x)
actually,the second question has two parts,the 1st is if function f is differentiable at any x,the 2nd is to prove f'(x)=f(0)*f(x)

Answer 1

a) take x=y=0
from the property f(x+y)=f(x)*f(y)
we get f(0+0)=f(0)^2
so that f(0)=0 or f(0)=1 QED
b)The question is incomplete.
f(x)=f(0+x)=f(0)*f(x)
if f(0)=0 then f(x)=0 and hence f is differentiable and f'(x)=0=f(0)*f(x)
for the case f(0)=1
f(dx-dx)=f(dx)f(-dx)=1
in the limit that dx->0, f(dx)=f(-dx)=1 so that f is differentiable at x=0
then, consider [f(x+dx)-f(x)]/dx=[f(x)f(dx)-f(x)]/dx=f(x)f'(0) in the limit dx->0, which is defined for all x since f(x) is defined and f'(0) is defined.
therefore f is differentiable.
and f'(x)=f'(0)f(x)
however is it not possible to prove that f'(x)=f(0)*f(x)
if f(x)=exp(x) it is true but if f(x)= a^x where a is any real number but e, it will not be true but all other properties of f are observed.