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Math Help - Peicewise Functions Continuity

  1. #1
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    Peicewise Functions Continuity

    I am doing some review, starting back at the fundamentals and I do not know why but piecewise confuse the hell out of me at times. I have the following piecewise function and have to tell if it is continuous or not on the interval [-1,1]

    f(x)=\frac{x}{|x|}  x not equal to 0

    f(x)= x = 0 when x = 0

    Sorry to be so abrupt, I know how to tell continuity with normal functions such as if it a rational fraction and long as the value that deems the function undefined is not in that interval it is continuous? also if the sign changes from left to right in a interval?

    Any tips to help me with continuity
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  2. #2
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    Quote Originally Posted by The Power View Post
    Any tips to help me with continuity
    Graph it out and make sure it doesn't have any gaps
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  3. #3
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    We are told to evaluate it algebraically since we are not allowed to use calculators on a test. Thanks for the help though, it know its not terrible hard, its just that piecewise functions to through me off at times.

    Edit: I think I finally recall my information and remove this mental block. A piecewise function is a whole function just has different rules for set intervals. So in sense if the interval for
    f(x)=\frac{x}{|x|} when x is not equal to 0,

    Sorry I forgot latex tags. So if the rule was a tad different and allowed 0, this would cause the function to be undefined causing a "gap" or hole in the graph thus rendering it no longer continuous, however since we have the rule when x equals 0 the output is 0 this allow the graph to continue on the interval [-1,1] Please someone correct me if I am wrong.
    Last edited by The Power; August 26th 2009 at 08:15 PM.
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  4. #4
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    \frac{x}{{\left| x \right|}} = \left\{ {\begin{array}{rl}<br />
   {1,} & {x > 0}  \\<br />
   { - 1,} & {x < 0}  \\<br /> <br />
 \end{array} } \right.
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  5. #5
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    As Plato said, if x< 0, |x|= -x so \frac{|x|}{x}= -1 for x< 1. If x> 0, |x|= x so \frac{|x|}{x}= 1 for x> 1.

    In order for the function f(x) to be continuous at x= a, three things must be true:
    1) f(a) is defined
    2) \lim_{x\rightarrow a} f(x) exists
    3) \lim_{x\rightarrow a} f(x)= f(a)
    (Since \lim_{x\rightarrow a} f(x)= f(a) pretty much implies the two sides exist, usuallly we just state (3).)

    Now, if \lim_{x\rightarrow a} f(x) exists then the two "onesided limits", \lim_{x\rightarrow a^-} f(x) and \lim_{x\rightarrow a^+} f(x) must exist and be equal. Think about what that means here.
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