Quantcast

Got Homework?

Connect with other students for help. It's a free community.

  • across
    MIT Grad Student
    Online now
  • laura*
    Helped 1,000 students
    Online now
  • Hero
    College Math Guru
    Online now

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

swissgirl

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 \( x \in \mathbb{R}\)

  • one year ago
  • one year ago

  • This Question is Closed
  1. satellite73
    Best Response
    You've already chosen the best response.
    Medals 0

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

    • one year ago
  2. swissgirl
    Best Response
    You've already chosen the best response.
    Medals 0

    Well you are proving backwards you gotta start from the beginning

    • one year ago
  3. satellite73
    Best Response
    You've already chosen the best response.
    Medals 0

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

    • one year ago
  4. satellite73
    Best Response
    You've already chosen the best response.
    Medals 0

    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?

    • one year ago
  5. swissgirl
    Best Response
    You've already chosen the best response.
    Medals 0

    Not exactly sure hmmmmm

    • one year ago
  6. satellite73
    Best Response
    You've already chosen the best response.
    Medals 0

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

    • one year ago
  7. swissgirl
    Best Response
    You've already chosen the best response.
    Medals 0

    Aha I see that

    • one year ago
  8. satellite73
    Best Response
    You've already chosen the best response.
    Medals 0

    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

    • one year ago
  9. swissgirl
    Best Response
    You've already chosen the best response.
    Medals 0

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

    • one year ago
  10. joemath314159
    Best Response
    You've already chosen the best response.
    Medals 2

    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.

    • one year ago
  11. swissgirl
    Best Response
    You've already chosen the best response.
    Medals 0

    Ohhh I like thisssss :)

    • one year ago
    • Attachments:

See more questions >>>

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.