Results 1 to 4 of 4

Math Help - epsilon-delta definition of continuity

  1. #1
    Member
    Joined
    Jan 2011
    Posts
    110

    epsilon-delta definition of continuity

    Just need to check the following, can anyone help?

    I need to use the epsilon-delta definition of continuity to prove that

    f(x) = x^2 - 4x

    is continuuous at 3

    This is what I have,

    The domain of f is R(the set of real numbers)

    Let epsilon>0 be given. We want to choose delta>0, in terms of epsilon, such that

    |f(x)-f(3)|<epsilon, for all x with |x-3|< delta (1.0)

    1. First we write
    f(x)-f(3) = x^2 - 4x + 3 = (x-1)(x-3)

    2. Next we obtain an upper bound for |x+3|, when x is near 3.
    If |x-3|<or equal to 1, than x lies in the closed interval [1,4], so
    |x+3|<or eqaul to|x|+3(Triangle Inequality)
    <or equal to 4+3=7

    3. Hence
    |f(x)-f(3)|<or equal to 7 |x-3|, for |x-3|<or equal to 1.

    So if |x-3|< delta, where 0<delta<or equal to 1, then
    |f(x)-f(3)|,7delta

    Now 7delta<or equal to epsilon if and only if delta < or equal to 1/7 epsilon. Thus if we choose delta = min {1,1/7epsilon}, then
    |f(x)-f(3)| < 7delta<or equal to 7 times 1/7 epsilon = epsilon, for all x with |x-3|< delta,

    which proves staement (1.0).

    Thus f is continuous at the point 3.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Siron's Avatar
    Joined
    Jul 2011
    From
    Norway
    Posts
    1,250
    Thanks
    20

    Re: epsilon-delta definition of continuity

    Indeed, choose |x-3|<1 \Leftrightarrow -1<x-3<1 \Leftrightarrow -1+2<x-3+2<1+2 \Leftrightarrow 1<x-1<3
    Therefore I would choose \delta=\min \{1,\frac{\epsilon}{3}\}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jan 2011
    Posts
    110

    Re: epsilon-delta definition of continuity

    Ok, thanks, make sense.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Siron's Avatar
    Joined
    Jul 2011
    From
    Norway
    Posts
    1,250
    Thanks
    20

    Re: epsilon-delta definition of continuity

    You're welcome!
    If you have more problems with epsilon-delta proofs then I recommand you to take a look at the first sticky thread of Krizalid (forum: calculus).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. epsilon delta definition
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 16th 2010, 09:26 AM
  2. epsilon delta-definition
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: August 27th 2009, 06:29 AM
  3. topology epsilon-delta definition of continuity
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: February 23rd 2009, 04:10 AM
  4. epsilon-delta definition of continuity
    Posted in the Calculus Forum
    Replies: 3
    Last Post: June 3rd 2008, 08:51 AM
  5. epsilon-delta definition
    Posted in the Calculus Forum
    Replies: 3
    Last Post: November 12th 2007, 10:45 AM

Search Tags


/mathhelpforum @mathhelpforum