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[prove] Continuous function

  1. Nov 27, 2008 #1
    1. The problem statement, all variables and given/known data

    Prove that the function:

    [tex]\frac{2x-1}{x^2+1}, x \in \mathbb{R}[/tex]

    is continuous.

    2. Relevant equations
    Definition 1.

    The function y=f(x) satisfied by the set Df is continuous in the point x=a only if:

    10 f(x) is defined in the point x=a i.e. [itex]a \in D_f[/itex]

    20 there is bound [tex]\lim_{x \rightarrow a}f(x)[/tex]

    30 [tex]\lim_{x \rightarrow a}f(x)=f(a)[/tex]

    Theorem 1.
    If the functions y=f(x) and y=g(x) are continuous in the point x=a Є Df ∩ Dg, then in the point x=a these functions are continuous:
    y=f(x)+g(x), y=f(x)g(x) and y=f(x)/g(x), if g(a) ≠ 0.

    3. The attempt at a solution

    I tried using the definition 1.

    But also this function is composition of two functions f(x) and g(x), so can I use the fact that f(x)=2x-1 and g(x)=x2+1 are continuous, and y=f(x)/g(x), g(a) ≠ 0 since x2+1 ≠ 0 ?
     
  2. jcsd
  3. Nov 27, 2008 #2
    Let f(x) = 2x - 1 and g(x) = x^2 + 1. Are f(x) and g(x) continuous functions? Is f(x)/g(x) continuous on it's domain?
     
  4. Nov 27, 2008 #3
    Yes, that's what I thought.
    But how will I prove for f(x)=x/(x+1), x Є R \ {-1} ?
     
  5. Nov 27, 2008 #4
    Do you know (or are allowed to use) the Algebraic Continuity Theorem?
     
  6. Nov 27, 2008 #5

    Mark44

    Staff: Mentor

    You can show that f(x) = x/(x + 1) satisfies all three of the conditions you listed in your first post. I.e., a) that f is defined at a (where a != -1, which is not in R \ {-1}), b) lim f(x) as x approaches a exists, and c) lim f(x) = f(a), as x approaches a.
     
  7. Nov 28, 2008 #6
    VeeEight could you please specify on what theorem do you mean? I am supposed to use the definition 1. or theorem 1. in the first post.
    10 f(a)= a/(a+1)

    20 [tex]\lim_{x \rightarrow a}f(x)=\lim_{x \rightarrow a}\frac{x}{x+1}=\frac{\lim_{x \rightarrow a}(x)}{\lim_{x \rightarrow a}(x+1)}=\frac{a}{a+1}[/tex]

    30 [tex]\lim_{x \rightarrow a}f(x)=\frac{a}{a+1}=f(a)[/tex]

    Should I prove the other tasks like this?

    Because I got:

    f(x)=sin(2x+3), x Є R

    and

    f(x)=ln(x-2), x Є R

    Thanks in advance.
     
    Last edited: Nov 28, 2008
  8. Nov 28, 2008 #7

    Mark44

    Staff: Mentor

    Yes, except for ln(x - 2), it must be that x > 2.
     
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