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!

Prove that f is continuous

  1. Oct 3, 2012 #1
    1. The problem statement, all variables and given/known data
    Suppose that f is an odd function satisfying [itex]\mathop {\lim }\limits_{x \to {0^ + }} f(x) = f(0)[/itex]. Prove that f(0)=0 and f is continuous at x=0.



    2. Relevant equations



    3. The attempt at a solution
    Since f is an odd function [tex]f(0) = - f(0) \Rightarrow f(0) = 0[/tex]
    Let t=-x, then when [itex]x \to {0^ + },t \to {0^ - }[/itex]
    [itex]\mathop {\lim }\limits_{x \to {0^ + }} f(x) = \mathop {\lim }\limits_{t \to {0^ - }} f( - t) = - \mathop {\lim }\limits_{t \to {0^ - }} f(t) = - \mathop {\lim }\limits_{x \to {0^ - }} f(x) = f(0) = 0[/itex]
    Therefore [itex]\mathop {\lim }\limits_{x \to {0^ + }} f(x) = \mathop {\lim }\limits_{x \to {0^ - }} f(x) = f(0)[/itex], which implies that f(x) is continuous at 0.

    Is my working correct?
     
  2. jcsd
  3. Oct 3, 2012 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Yes, it is.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Prove that f is continuous
Loading...