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Limits and Continuity

  1. Sep 13, 2009 #1
    I am reading through calc1 and reviewing Limits and Continuity/Discontinuity, I have so many questions!

    There is a theorem here, it says if F and G are continuous at A and C is a constant, then the following are continues at a: F+G, F-G, CF, FG, F/G (G!=0)
    however the explanation I read for this makes no sense to me! (F+G)(A)

    Another question I have, for continuous functions, say I am given a piecewise function:
    cx^2+2x if x<2
    x^3 - cx if x>= 2
    I am suppose to find what makes this function continuous everywhere. I went back to read my book to find an example of this to break it down to the most simplest steps but could not find anything T.T (Squeezes teacher, I must solve). How do I start to even begin this problem?

    I was looking at a graph, it was just a random function with squiggly lines. Several breaks in the function was no problem, I noticed that as my eye went from left to right, in one spot the line broke, a dot appeared over it, and then the line started in the same direction from a different starting point. Since the little dot randomly appeared over the break, does that mean that the point at which the line broke was moved? or is it a whole different function?

    O_O ahh brain burnz, I like cheese.
     
  2. jcsd
  3. Sep 13, 2009 #2

    mathman

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    Your first question is too broad. It looks like you need to understand what continuity means.

    For the second question:
    At x=2, the x<2 expression is 4c+4, while the x>2 expression is 8-2c. For continuity, you need to find the value of c where they are equal:
    4c+4=8-2c.
     
  4. Sep 13, 2009 #3
    All of these statements are direct consequences of the corresponding basic limit theorems, since continuity is defined by the existence of a limit equal to the value of the function at a point. Review your chapter on limits.
     
  5. Sep 14, 2009 #4

    HallsofIvy

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    What was the explanation then? The only proof I've ever seen is:

    To prove that F+ G is constant at x= A, look at
    [tex]\lim_{x\to A} F(x)+ G(x)[/tex]
    by the basic limit theorem for sums, that is
    [tex]\lim_{x\to A}F(x)+ \lim_{x\to A}G(x)[/tex]
    and, since F is continuous at A,
    [tex]\lim_{x\to A}F(x)= F(A)[/tex]
    (that is the definition of "continuous at x= A")
    and, since G is continuous at A,
    [tex]\lim_{x\to A}G(x)= G(A)[/tex]
    so that
    [tex]\lim_{x\to A}F(x)+ G(x)= F(A)+ G(A)[/tex]
    proving that F+ G is continous at x= A.
    What does not make sense about that?

    For x< 2 f(x)= [itex]cx^2+ 2x[/itex] so
    [tex]\lim_{x\to 2^-} f(x)= \lim_{x\to 2^-} cx^2+ 2x= 4c+ 4[/itex]
    because [itex]cx^2+ 2x[/itex] is a polynomial and so always continuous.

    For x> 2, f(x)= [itex]x^3- cx[/itex] so
    [tex]\lim_{x\to 2^+} f(x)= \lim_{x\to 2^+} x^2- cx= 8- 2c[/itex]
    again because [itex]x^3- cx[/itex] is a polynomial.

    In order to be continuous at x= 2 we must first have that f(x) has a limit at x= 2 which means that those two one-sided limits must be the same: [itex]4c+4= 8- 2c[/itex].

    Solve that equation for c. We must also have that the limit is equal to the value of the function there so, whatever you get for c, set f(2) equal to the common value of 4c+4 and 8- 2c.
     
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