Results 1 to 4 of 4

Math Help - Continuity

  1. #1
    Member roshanhero's Avatar
    Joined
    Aug 2008
    Posts
    180

    Continuity

    1.Using \varepsilon-\delta definition,prove that f(x)=cosx is continuous.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,560
    Thanks
    1425
    Quote Originally Posted by roshanhero View Post
    1.Using \varepsilon-\delta definition,prove that f(x)=cosx is continuous.
    What would using the \epsilon - \delta definition achieve.

    It's pretty obvious from definition of cosine as the horizontal distance made from the unit circle that \cos{x} is continuous for all possible angles x...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member roshanhero's Avatar
    Joined
    Aug 2008
    Posts
    180
    I want to prove it from the formal definition as well.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,654
    Thanks
    13
    Then if you want to prove it then do it.

    I'll do it for you because I want to do it.

    -----

    It's easy to prove that |\sin x-\sin y|\le|x-y| for each x,y\in\mathbb R, in particular put y=0 and we get this useful inequality |\sin x|\le|x|. As for the sake of the problem given \epsilon>0, then for 0<\delta=\epsilon, we have

    \left| \cos (x)-\cos (a) \right|=\left| 2\sin \left( \frac{x+a}{2} \right)\sin \left( \frac{x-a}{2} \right) \right|\le 2\left| \sin \left( \frac{x-a}{2} \right) \right|, hence |\cos (x)-\cos(a)|=|x-a|<\epsilon.

    As required.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. continuity
    Posted in the Calculus Forum
    Replies: 5
    Last Post: April 20th 2011, 01:36 PM
  2. y=x^3 (continuity)
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 16th 2010, 10:50 PM
  3. continuity
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: April 9th 2010, 04:23 PM
  4. Continuity
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 7th 2009, 09:10 PM
  5. Continuity, Uniform Continuity
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 1st 2009, 08:36 PM

Search Tags


/mathhelpforum @mathhelpforum