More About Continuity

Discussion in 'Calculus' started by nycmathguy, May 13, 2022.

  1. nycmathguy

    nycmathguy

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    Calculus
    Section 2.5

    Define the following in your own words and give an example for each.

    1. Removable discontinuity

    2. Infinite discontinuity

    3. Jump discontinuity
     
    nycmathguy, May 13, 2022
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  2. nycmathguy

    MathLover1

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    A function is continuous when its graph is a single unbroken curve that you could draw without lifting your pen from the paper.

    Here is a continuous function:
    upload_2022-5-13_14-27-37.jpeg
    When a function is continuous within its Domain, it is a continuous function.


    So what is not continuous (also called discontinuous) ?

    upload_2022-5-13_14-34-37.jpeg

    Look out for holes, jumps or vertical asymptotes (where the function heads up/down towards infinity).

    Example: 1/(x−1)
    At x=1 we have:

    1/(x−1) = 1/0 = undefined

    So there is a "discontinuity" at x=1
    [​IMG]

    So f(x) = 1/(x−1) over all Real Numbers is NOT continuous

    Let's change the domain to x>1

    [​IMG]

    So g(x) IS continuous

    In other words g(x) does not include the value x=1, so it is continuous.

    We can define continuous using Limits:
    A function f is continuous when, for every value c in its Domain:

    f(c) is defined, and lim f(x) = f(c) as x→c
    "the limit of f(x) as x approaches c equals f(c)"

    The limit says:

    "as x gets closer and closer to c
    then f(x) gets closer and closer to f(c)"

    And we have to check from both directions:
    [​IMG] as x approaches c (from left)
    then f(x) approaches f(c)

    and

    [​IMG] as x approaches c (from right) then f(x) approaches f(c)


    If we get different values from left and right (a "jump"), then the limit does not exist!
    And remember this has to be true for every value c in the domain.



    1. Removable discontinuity

    What Are Holes?
    Another way we can get a removable discontinuity is when the function has a hole. A hole is created when the function has the same factor in both the numerator and denominator. This factor can be canceled out but needs to still be considered when evaluating the function, such as when graphing or finding the range. When dealing with a function like this, there will be some point where the function is undefined. Look at this function, for example.

    f(x)=(x^2-4x)/(x-4)........factor numerator
    f(x)=(x(x-4))/(x-4)
    This function has the factor x - 4 in both the numerator and denominator. What happens at the point x = 4? Let's see.

    f(x)=(4(4-4))/(4-4)=(4*0)/0 =0/0 =>the function is undefined

    In the graphs below, there is a hole in the function at x=a. These holes are called removable discontinuities.


    2. Infinite discontinuity

    If the function doesn't approach a particular finite value, the limit does not exist. This is an infinite discontinuity.

    The following two graphs are also examples of infinite discontinuities at x=a. Notice that in all three cases, both of the one-sided limits are infinite.
    [​IMG]

    3. Jump discontinuity
    The function is approaching different values depending on the direction x is coming from. When this happens, we say the function has a jump discontinuity at x=a.

    The graph of f(x) below shows a function that is discontinuous at x=a.
    [​IMG]
    In this graph, you can easily see that lim(−f(x))=L as x→a and lim(+f(x))=M as x→a.
     
    MathLover1, May 13, 2022
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    nycmathguy likes this.
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  3. nycmathguy

    nycmathguy

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    Wow! You outdid yourself. Best reply ever.
     
    nycmathguy, May 13, 2022
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    MathLover1 likes this.
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