Question

Assume \( f: \mathbb{R} \to \mathbb{R}\) is such that \(f(x+y)=f(x)f(y)\) ( The class of exponential functions has this property). Prove that f having a limit at 0 implies that f has a limit at every real number and is one, or f is identically 0 for every \( x \in \mathbb{R}\)

Answer 1

if \(f\) is identically zero there is nothing to prove

Answer 2

Well you are proving backwards you gotta start from the beginning

Answer 3

i guess we have to worry about \(f\) being continuous
start by showing \(f(0)=1\)

Answer 4

this line
Prove that f having a limit at 0 implies that f has a limit at every real number and is one, does not really make sense to me
we can show that \(f(0)=1\) but for example if \(f(x)=e^x\) then what does "\(f\) has a limit at every real number and is one" mean?

Answer 5

Not exactly sure hmmmmm

Answer 6

\[f(x)=f(x+0)=f(x)f(0)\implies f(0)=1\]

Answer 7

Aha I see that

Answer 8

if \(f\) is continuous at \(0\) then
\[\lim_{h\to 0}f(x+h)=\lim_{h\to 0}f(x)f(h)=f(x)\] making \(f\) continuous

Answer 9

Thanks :) Just gonna string some stuff together here :)

Answer 10

you could write the hypothesis as a limit, then do a change of variable. Since the limit as x goes to 0 exists, we have:\[\lim_{x\rightarrow 0}f(x)=c\]for some c. Let b be a fixed real number, and let x = y-b. Then saying x goes to zero is equivalent to saying y goes to b. So we get this now:\[\lim_{y\rightarrow b}f(y-b)=c\Longrightarrow \lim_{y\rightarrow b}f(y)(f(-b)=c\]\[\Longrightarrow f(-b)\lim_{y\rightarrow b}f(y)=c\]Note that since f(0)=1, it follows that:\[1=f(0)=f(x+(-x))=f(x)f(-x)\]for all x. Therefore\[f(-b)\lim_{y\rightarrow b}f(y)=c\Longrightarrow \lim_{y\rightarrow b}f(y)=cf(b)\]So the limit exists at b. Since b was arbitrary, the limit exists everywhere.

Answer 11

Ohhh I like thisssss :)