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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

    Dick

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    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

    Dick

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    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.
     
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