Question on the functional equation $f(x+y)=f(x)f(y)$

Question

Are there any functions other than $a^{kx}$ that satisfy the relation $$f(x+y)=f(x)\cdot f(y)\text?$$

Actually I have a question that states that there exists a function such that $f(x+y)=f(x)f(y)$ and $f(5) =2$, $f'(0)=3$, then $f'(5)=$?
I thought of assuming the function as $a^{kx}$ and proceeded but I don't get my solution right.

http://math.stackexchange.com/questions/423492/overview-of-basic-facts-about-cauchy-functional-equation — Surb, Jan 06 '17 at 17:33
If you add the condition that f be continuous then the answer is "no". — user247327, Jan 06 '17 at 17:34

Damian Sowinski · Answer 1 · 2020-12-29T16:31:05.820

2

There is no need to solve this equation. Just compute: $$ \frac{\partial}{\partial x} f(x+y) = f'(x+y) $$ and $$ \frac{\partial}{\partial x} \left[f(x)f(y)\right] = f'(x)\cdot f(y) $$ Equate these two and set $x=0$, $y = 5$ to find: $$ f'(5) = f'(0)\cdot f(5) = 3\cdot 2 = 6 $$

edited Dec 29 '20 at 16:31

answered Jan 06 '17 at 17:48

Damian Sowinski

235

Why did you put f'(y)=0? And why is my method incorrect when I have enough conditions to generate the function itself? – Piyush Kumar Jan 06 '17 at 17:50
I didn't do that. Reread my response. Your method would work if you did all the algebra correctly, but it is overkill. – Damian Sowinski Jan 06 '17 at 18:09

Question on the functional equation $f(x+y)=f(x)f(y)$

1 Answers1