As the question states, show that the property exhibited can only be satisfied by a logarithmic function i.e no other family of functions can satisfy the above property.
Answer
Continuity is necessary.
If F(x+y)=F(x)+F(y), for all x,y and F discontinuous (such F exist due to the Axiom of Choice, and in particular, the fact that R over Q possesses a Hamel basis) and f(x)=F(logx), then f(xy)=f(x)+f(y), and f is not logarithmic!
No comments:
Post a Comment