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Math Help - Continuous and discontinuous functions

  1. #1
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    Continuous and discontinuous functions

    Hello,

    I'm having trouble understanding continuous and discontinuous functions. I understand the basic concept of continuous functions are functions where small changes in input result in small changes in output, but isn't that the case with all functions? I understand them as far as graphing goes, but I'm not fully understanding the overall concept.

    Why is y=1/x discontinuous but y=x*2 continuous?

    Can I get a simple explanation?

    Thanks.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by SeanC View Post
    Hello,

    I'm having trouble understanding continuous and discontinuous functions. I understand the basic concept of continuous functions are functions where small changes in input result in small changes in output, but isn't that the case with all functions? I understand them as far as graphing goes, but I'm not fully understanding the overall concept.

    Why is y=1/x discontinuous but y=x*2 continuous?

    Can I get a simple explanation?

    Thanks.
    Look at a graph of y=1/x near x=0, you will see that it goes to +infty as you
    approch 0 from above and -infty as you approach 0 from below, so near 0
    a small change in x cn result in a large change in y, hence this is
    discontinuous at x=0.

    For you other example if you change x to x+e, y changes from x*2 to
    x*2+e*2, and as e is small so is e*2.

    Note by convention in ASCII (plain text) * denotes multiplication. If you
    meant x squared you should use x^2. In this case y changes from x^2 to
    (x+e)^2 = x^2 + 2e x + e^2, and if e is small enough 2e x+e^2 is also
    small.

    RonL
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  3. #3
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    The actual definition of a continous function is as follows. Vageuly as I remember it from 1st and 2nd year undergrad.

    a function f(x) is continous at a point a if

    \lim_{\epsilon \rightarrow 0} f(a-\epsilon)=\lim_{\epsilon \rightarrow 0}f(a+\epsilon).

    A function is continous everwhere if it is continous for all a. So take the function
    f(x)=\frac{1}{x}

    at x=0.

    \lim_{\epsilon \rightarrow 0} (\frac{1}{-\epsilon}) \rightarrow -\infty

    \lim_{\epsilon \rightarrow 0}(\frac{1}{\epsilon}) \rightarrow +\infty

    The two aren't equal, so the function is dicontinous at x=0



    The simple definition is...

    A function is continous if you can draw it WITHOUT lifting pen off paper.
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