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}$

This is what I have so far. I know we can demonstrate this using the exponential function where $$f(0)=a^0=1$$ Hence $$f(x+0)=f(x)f(0)=a^x*a^0=a^x=f(x)$$ 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.

Can someone please guide me?

  • $\begingroup$ Exactly the same. Forget the $a$ $\endgroup$ – leo Feb 1 '13 at 4:05
  • $\begingroup$ What? I don't follow what you are trying to say $\endgroup$ – math101 Feb 1 '13 at 4:08
  • $\begingroup$ Like how would I continue this? $\endgroup$ – math101 Feb 1 '13 at 4:09
  • 3
    $\begingroup$ You don't actually have to assume it's an exponential function, so you can remove all the parts that use $a^x$. $\endgroup$ – Robert Mastragostino Feb 1 '13 at 4:09
  • $\begingroup$ Ohhh I get it. Thanks :) $\endgroup$ – math101 Feb 1 '13 at 4:10

For the second, set $y=0$ and get $\forall xf(x)=f(x)f(0)$. If there is any number $x$ such that $f(x) \ne 0$ we get that $f(0)=1$

  • $\begingroup$ Isn't that quite similar to the method I have used? $\endgroup$ – math101 Feb 1 '13 at 4:38
  • $\begingroup$ @math101: yes, but it seemed you were asking about it in your last comment "how would I prove the second part". Maybe I got the wrong second part. $\endgroup$ – Ross Millikan Feb 1 '13 at 4:47
  • $\begingroup$ What does it mean that f is identically $0$ for $ x\in \mathbb{R}$? $\endgroup$ – math101 Feb 1 '13 at 4:50
  • $\begingroup$ @math101: that is a perfectly good function. No matter what $x$ you feed it, $f(x)=0$. In this proof, it means I can't find an $x$ so I can divide by $f(x)$, so the proof that $f(0)=1$ fails. A good thing, because $f(0)=0$. $\endgroup$ – Ross Millikan Feb 1 '13 at 5:02

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