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!

Proof that a function is continuous on its domain

  1. Apr 30, 2013 #1
    1. The problem statement, all variables and given/known data

    We have [itex]f(x) = \frac{x^{2}+x-2}{x-1}+cos(x) , x\in\mathbb{R}\setminus \{1\}[/itex] and wish to prove that it is continuous on its domain.

    2. Relevant equations

    The delta-epsilon definition of the continuity of a function.

    3. The attempt at a solution

    I've managed to reduce [itex]|f(x) - f(x_0)| [/itex]to[itex] |x-x_0| + |cos(x) - cos(x_0)| < \delta + |cos(x) - cos(x_0)|[/itex]
    I'm not too sure where to go from there or even if I'm on the right track. Any insight would be greatly appreciated.
  2. jcsd
  3. May 1, 2013 #2


    User Avatar
    Science Advisor
    Homework Helper

    Factoring x^2+x-2 would be a great first step.
  4. May 1, 2013 #3
    I have, that's how I arrived at the reduced expression. The questions is where to go from there. I could always use the property that if [itex]f[/itex] and [itex]g[/itex] are continuous at a point [itex]x_0\in \mathbb{A}[/itex] then [itex]f+g[/itex] is continuous at [itex]x_0[/itex]. But I don't know how to prove that [itex]cos(x)[/itex] is continuous on the domain.
  5. May 1, 2013 #4


    User Avatar
    Science Advisor
    Homework Helper

    Yes, that you did. One way to prove cos is continuous is to use the trig identity, cos(u)-cos(v)=(-2)sin((u+v)/2)*sin((u-v)/2).
  6. May 1, 2013 #5
    Ahh I overlooked that, thanks! I think I've got it now.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted