A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

Loser66

  • one year ago

If f(x) is differentiable at x0. Prove it continuous at x0 Please, help

  • This Question is Closed
  1. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @xapproachesinfinity hey kid, help old man please

  2. xapproachesinfinity
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    hey old man long time

  3. xapproachesinfinity
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    i think theorem state if f is differentiable applies f is continuous so you are proving the theorem

  4. xapproachesinfinity
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    so let see f differentiable at x0 means the limits exist f'(x0)

  5. xapproachesinfinity
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    so \(\lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}=\text{exist}\)

  6. xapproachesinfinity
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    you have to work from here

  7. xapproachesinfinity
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    first let's state the definition of f being cont at x0 that is to say \(\lim_{x\to x_0} f(x)=f(x_0)\)

  8. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Sorry, kid, old man was kicked out of the site.

  9. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    But I think you got circulation argument: differentiable --> continuous --

  10. Loser66
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    However, I got it. \(f(x) - f(x_0) = \dfrac{f(x) -f(x_0)}{x-x_0}(x-x_0)\) \(lim_{x\rightarrow x_0} |f(x) - f(x_0)| = lim_{x\rightarrow x_0}\dfrac{|f(x) -f(x_0)}{|x-x_0|}|x-x_0|\) since f(x) differentiable at \(x_0\), the limit exists and as x approaches \(x_0\) \(|x-x_0|=0\) That gives us \(lim_{x\rightarrow x_0}|f(x) -f(x_0)|=0\) or \(lim_{x\rightarrow x_0}f(x) = f(x_0)\) That shows f(x) continuous at x0

  11. xapproachesinfinity
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    there is no circulation it is one way f different implies f continuous so you start with f different stating the definition limf(x)-f(x0)/x-x0=f'(x0) multilply both sides by (x-x0) we get lim(f(x)-f(x0)=f'(x0)(x-x0) as x tends to x0 the quanties (x-x0) shrinks to zero (limit concept) then we are safe to say lim(f(x)-f(x0)=f'(x0)0=0 therefore lim(f(x))=f(x0) hence f is continuous at x0

  12. xapproachesinfinity
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    a subtlety here is the |x-c| is zero but it is not zero however we write as it is zero

  13. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

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.