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Continuous Functions

  1. Jun 18, 2016 #1
    1. The problem statement, all variables and given/known data
    The problem is posted below in the picture. I looked at c and d and can do those. I am unsure about a and b.

    2. Relevant equations


    3. The attempt at a solution
    I looked at graphing the problems, but I think it is a wrong approach.
     

    Attached Files:

  2. jcsd
  3. Jun 18, 2016 #2

    Mark44

    Staff: Mentor

    Graphing a) and b) might be helpful. What is the definition of continuity at a point?
     
  4. Jun 19, 2016 #3
    Continuity at a point: A function f is continuous at c if the following three conditions are met.

    f(C) is defined

    lim as x approaches c exists.

    lim as x approaches c of f(x) = f(c)
     
  5. Jun 19, 2016 #4

    Mark44

    Staff: Mentor

    You said you were unsure about parts a and b. What are your thoughts about these problems so far?
     
  6. Jun 19, 2016 #5
    For "a" I was thinking it cant be continuous because the graph would be broken up and you technically could not draw the graph without picking up your pencil.

    For "b" I was thinking 0, but I think it is wrong.
     
  7. Jun 19, 2016 #6

    Mark44

    Staff: Mentor

    For a) why is |f| continuous? For b) you say, "I was thinking 0." Can you elaborate on this? Try to be as clear as can about what you mean.

    In your class has the definition of the limit been given yet? By that I mean the ##\delta - \epsilon## definition?
     
  8. Jun 19, 2016 #7
    Attached is the definition about limits.

    I think "a" is not continuous because the graph will be alternating between 1 and -1 and not stay at a straight line. I would have to pick up my pencil to draw the graph.

    For "b" I was thinking 0, because the limit as s approaches 0, the graph approaches 0.
     

    Attached Files:

  9. Jun 19, 2016 #8

    Mark44

    Staff: Mentor

    That's not what I asked about -- I asked what about |f|? Is it continuous?
    Also, since you have the limit definition to work with, why isn't f continuous anywhere? Can you say this in some way more rigorous than "picking up the pencil to draw the graph"?
    The problem doesn't involve s. Can you write something in more mathematical terms? I.e.,
    $$\lim_{x \to 0} f(x) = ?$$
    I used LaTeX to write this. In unrendered form this is $$\lim_{x \to 0} f(x) = ?$$
     
  10. Jun 19, 2016 #9
    For "b" I was thinking 0, because when you look at the limit you see the following.
    $$\lim_{x \to 0} f(x) = 0$$
    So the function would be defined at that point.


    For "a" I was thinking the absolute value function would be continuous, because I am thinking the -1 will change to positive 1.
     
  11. Jun 19, 2016 #10

    Mark44

    Staff: Mentor

    All of the functions listed in parts a through d are defined for all real numbers, so that's not the issue. What are they asking you in this problem?
    The absolute value of the function, or |f|, is the one that is continuous. If you are given a value of ##\epsilon## would you be able to find a number ##\delta > 0## for which ##| |f(x)| - 0| < \epsilon## when ##|x - 0| < \delta##?
     
  12. Jun 19, 2016 #11
    They are asking me to show f(x) is nowhere continuous. Well it is nowhere continuous because it is not continuous at each point in the interval. I think I got a now. would this work?

    upload_2016-6-19_13-29-30.png

    Is this an idea of what you are getting at.

    ##\epsilon## = 1/2
    ##\delta > 0## = 1/4
    ##| |f(1/8)| - 0| < \epsilon##
    when ##|1/8 - 0| < \delta##?
     

    Attached Files:

    Last edited by a moderator: Jun 19, 2016
  13. Jun 19, 2016 #12

    Mark44

    Staff: Mentor

    More like the example you posted as an image, the one with the two sequences.
     
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