is my solution to this "oxfords admissions interview" valid?

[dire tusk]

the guy in the yo8utube video spent 20 minutes rigorously doing limits and everything crazy, my solution was very simple and i wanna check theres no fault:

f(x+y) = f(x)f(y)
immediately think of indicies:
a^(x+y) = a^x * a^y
so we know f(x) = a^x
f'(x) = a^x * ln(a)
sub in 0:
f'(0) = a^0 * ln(a) = 3
1 * ln(a) = 3
a = e^3

thererore f(x) = e^(3x)

[wide comet]

you haven't proved that e^(3x) is the only solution

[dire tusk]

what other function has the property f(x+y) = f(x)f(y)

oh actually f(x) = 0 is oine

[wide comet]

see https://en.wikipedia.org/wiki/Cauchy's_functional_equation, it is isomorphic to this

there are infinitely many functions satisfying it that are not of the form a^x

however they are discontinuous everywhere

it shouldn't take 20 minutes to solve this problem, exactly

you can show there is some a such that f(x) = a^x for any rational number x

and then since it is differentiable it is continuous and the rational numbers are dense so blah