1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Would you use Trichotomy?

  1. Nov 3, 2008 #1
    Let [tex] X [/tex] be a metric space, let [tex] a \in X [/tex] be a limit point of [tex] X [/tex], and let [tex] f: X \to \mathbb{R} [/tex] be a function. Assume that the limit of [tex] f [/tex] exists at [tex] a [/tex]. Fix [tex] t \in \mathbb{R} [/tex]. Suppose there exists [tex] r > 0 [/tex] such that [tex] f(x) \geq t [/tex] for every [tex] x \in B_{r}(a) \backslash \{a \} [/tex]; then [tex] \lim_{x \to a} f(x) \geq t [/tex].

    How would you prove this? Would you use Trichotomy?
  2. jcsd
  3. Nov 3, 2008 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Re: Continuity

    Find a sequence xk such that xk converges to a. Then what can you say about f(xk) for each k?
  4. Nov 3, 2008 #3
    Re: Continuity

    you will get a contradiction pretty easy if lim f(x)<t.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook