1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Continuous function.

  1. Apr 19, 2004 #1
    Im having a little trouble with this question.

    If f is continuous at [tex]c[/tex] and [tex]f(c) < 5[/tex], prove that there exists a [tex]\delta > 0[/tex] such that [tex]f(x) < 7[/tex] for all [tex]x \in (c - \delta , c + \delta)[/tex]

    So we are given that f is continuous at c.
    So [tex]\lim_{x \to c}f(x) = f(c) < 5[/tex]
    [tex]\forall \epsilon > 0 \exists \delta > 0[/tex] such that whenever [tex]|x - c| < \delta[/tex] then [tex]|f(x) - f(c)| < \epsilon[/tex]

    [tex]|x - c| < \delta[/tex]
    [tex]-\delta < x - c < \delta[/tex]
    [tex]c - \delta < x < c + \delta[/tex]

    Ok now im getting lost..
    I know i have to do something with [tex]|f(x) - f(c)| < \epsilon[/tex]
    [tex]|f(x) - 5| < \epsilon[/tex]
    [tex]-\epsilon < f(x) - 5 < \epsilon[/tex]
    [tex]5 - \epsilon < f(x) < 5 + \epsilon[/tex]
    we want [tex]f(x) < 7[/tex] so .... am i on the right track??? How can i find the [tex]\delta > 0[/tex] that satisfies this?
  2. jcsd
  3. Apr 19, 2004 #2
    Yes, you're on the right track. Take [itex]\epsilon=2[/itex]. From the limit you mentioned, we know that there exists a [itex]\delta>0[/itex] such that [itex]3<f(x)<7[/itex] and so you are done. The question was asking you to show that a delta exists, and the limit demonstrates that it clearly does, so you are done.
  4. Apr 19, 2004 #3
    I was actually thinking about making [tex]\epsilon = 2[/tex] but thought it was to easy. Ok so thanks for explaining the question a bit more to me, i understand it now. :smile:
  5. Apr 19, 2004 #4

    Isn't what you're asked to prove equivalent to,

    [tex]\lim_{x \to c}f(x) = f(c) < 7[/tex]

    which follows trivially from,

    [tex]\lim_{x \to c}f(x) = f(c) < 5[/tex]

    which was what you were given?
  6. Apr 20, 2004 #5


    User Avatar
    Staff Emeritus
    Science Advisor

    No, that's not what was given. Given that the limit is less than 5 it is true, but requires proof, that, for x close to c, f(x)< 5. gimpy was asked to prove the slightly simpler case: that, for x close to c, f(x)< 7.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Continuous function.
  1. Continuous Function (Replies: 5)

  2. Continuous function? (Replies: 6)

  3. Continuous functions (Replies: 3)

  4. Continuous function (Replies: 8)