Pre-Calculus For Dummies, 3rd Edition
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A graph for a function that's smooth without any holes, jumps, or asymptotes is called continuous. Your pre-calculus teacher will tell you that three things have to be true for a function to be continuous at some value c in its domain:
  1. f(c) must be defined. The function must exist at an x value (c), which means you can't have a hole in the function (such as a 0 in the denominator).

  2. The limit of the function as x approaches the value c must exist. The left and right limits must be the same; in other words, the function can't jump or have an asymptote. The mathematical way to say this is that

    image0.png

    must exist.

  3. The function's value at c and the limit as x approaches c must be the same.

    image1.png
For example, you can show that the function

image2.png

is continuous at x = 4 because of the following facts:

  • f(4) exists. You can substitute 4 into this function to get an answer: 8.

    image3.png

    If you look at the function algebraically, it factors to this:

    image4.png

    Nothing cancels, but you can still plug in 4 to get

    image5.png

    which is 8.

    image6.png

    Both sides of the equation are 8, so f(x) is continuous at x = 4.

If any of the above situations aren't true, the function is discontinuous at that value for x.

Functions that aren't continuous at an x value either have a removable discontinuity (a hole in the graph of the function) or a nonremovable discontinuity (such as a jump or an asymptote in the graph):

  • If the function factors and the bottom term cancels, the discontinuity at the x-value for which the denominator was zero is removable, so the graph has a hole in it.

    For example, this function factors as shown:

    image0.png

    After canceling, it leaves you with x – 7. Therefore x + 3 = 0 (or x = –3) is a removable discontinuity — the graph has a hole, like you see in Figure a.

    The graph of a removable discontinuity leaves you feeling empty, whereas a graph of a nonremovable

    The graph of a removable discontinuity leaves you feeling empty, whereas a graph of a nonremovable discontinuity leaves you feeling jumpy.
  • If a term doesn't cancel, the discontinuity at this x value corresponding to this term for which the denominator is zero is nonremovable, and the graph has a vertical asymptote.

    The following function factors as shown:

    image2.png

    Because the x + 1 cancels, you have a removable discontinuity at x = –1 (you'd see a hole in the graph there, not an asymptote). But the x – 6 didn't cancel in the denominator, so you have a nonremovable discontinuity at x = 6. This discontinuity creates a vertical asymptote in the graph at x = 6. Figure b shows the graph of g(x).

About This Article

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About the book author:

Mary Jane Sterling aught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies.

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