Suppose that F R R is continuous and has the property that

Suppose that F : R ---> R is continuous, and has the property that F(x + y) = F(x)F(y) for all x and y in R. Show that either F(x) = 0 for all x in R or there exists b > 0 such that F(x) = bX for all x in R.

Solution

f(0+0)=f(0)=f(0)^2

Hence, f(0)=0 or 1

Case 1:f(0)=0

f(x+0)=f(x)=f(x)f(0)=0

for all x in R ,f(x)=0

Case 2:f(0)=1

f(x-x)=f(0)=1=f(x)f(-x)

f(x+x)=f(2x)=f(x)f(x)=f(x)^2

f(2x+x)=f(3x)=f(2x)f(x)=f(x)^3

f(nx)=f(x)^n

So, f(x) is an exponential function ie a^x for some a>0 in R