Results 1 to 4 of 4

Math Help - Continuity Question

  1. #1
    Member
    Joined
    May 2006
    Posts
    148
    Thanks
    1

    Continuity Question

    Can a function be continuous everywhere but differentiable nowhere?

    Also, why is the sum of infinite continuous functions a discontinuous one? Kind of crazy to me.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by fifthrapiers View Post
    Can a function be continuous everywhere but differentiable nowhere?
    Yes! (In fact I never go though a week without mentioning it at least once to somebody).

    Weierstraß.

    Also, why is the sum of infinite continuous functions a discontinuous one? Kind of crazy to me.
    Why not? We never proven that otherwise.

    (Why is a sum of infinite rational numbers rational (further more transendental!!!) same idea).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Apr 2006
    Posts
    401
    Quote Originally Posted by fifthrapiers View Post
    Can a function be continuous everywhere but differentiable nowhere?

    Also, why is the sum of infinite continuous functions a discontinuous one? Kind of crazy to me.
    For your first question, indeed! I'll even challenge you more. Did you know that a function can be continuous at a POINT- think about rational numbers? Also, fractals are nicely applied here.

    For the second question, there exists a function that is discontinuous when adding an infinite number of continuous functions (this is not true for all functions obviously). There's even a proof in complex analysis that shows an infinite number of continuous functions being added together to create a discontinuous one.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by AfterShock View Post
    There's even a proof in complex analysis that shows an infinite number of continuous functions being added together to create a discontinuous one.
    Why be so fancy to use complex analysis?

    Define the sequence of functions,

    f_n(x) = x^n on [0,1]

    We can see that,

    lim f_n(x) = f(x) (pointwise) where,

    f(x) = 0 if x in [0,1) and 1 if x =1.

    Note the sequence of functions {f_n(x)} are continous on [0,1].
    But their pointwise limit f(x) is not on [0,1].
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Question regarding continuity
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 29th 2011, 08:20 AM
  2. Continuity Question
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: January 22nd 2011, 03:04 AM
  3. Question regarding continuity
    Posted in the Calculus Forum
    Replies: 7
    Last Post: February 18th 2010, 09:49 AM
  4. Continuity question
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: July 3rd 2009, 10:10 AM
  5. continuity question
    Posted in the Calculus Forum
    Replies: 1
    Last Post: August 20th 2008, 02:55 AM

Search Tags


/mathhelpforum @mathhelpforum