Ok, I know how you feel about homework threads… I’m not looking for

answers, I’m looking for a direction here.

My calc I teacher gave us this project to do. We’re given some

information, and we have to do some things with it, in 3 separate

mini-projects. Here’s what I have, and my ideas on it.

**Information**

1: f(a+b)=f(a)f(b)

2: f(0)=0

3: f(x) is differentiable at x=0, and f(0)=1

Regardless of my response to project 1, the only things I know about a

function f(x) are properties 1-3, IOW, my answer for project 1 doesn’t

matter when doing the other two projects.

**Projects**

1: Find a function f(x) that satisfies all of the above conditions

So I think I’ve succeded here, with a simple f(x)=e[sup]x[/sup]

function. This isn’t where I’m having the problems.

2: Show that condition 1 implies that either f(0)=0 or f(0)=1. Then

show that if f(0)=0 then f(x)=0 for all x, while if f(0)=1 then f(x)

!=0 for all x.

Huh? Here, the only thing I’ve been able to think of that might work

would be the Mean Value Theorem, using f(0)=0 for my first point, but

I don’t have a 2nd point to define the interval on. Is this even the

right direction to go here? Would saying something like f(0+0)=2f(0)

help there?

- Using the limit definition of a derivative, show that

f’(x)=f(x)f’(0) for all x. What does this say about the continuity of

f(x) for all x?

I think that here I need to use the product rule, by saying

f(x+0)=f(x)f(0), and then using the product rule to say that

f’(x+0)=f’(x)f(0)+f(x)f’(0), and since information point 2 says that

f(0)=0, then the first term goes away. I think I’m on the right track

there, but I don’t know what to do to say anything about the

continuity. Is it as simple as saying that since f(x) is

differentiable over all x, then it’s continuous over all x?