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Homework Help: Is a graph Continuous and differentiable at a given point

  1. Mar 8, 2009 #1
    1. The problem statement, all variables and given/known data

    F
    f(x)={(2x-1)/Absolute value(2x-1) x cannot equal (1/2)
    { 0 x = (1/2)

    a) is f continous at X = (1/2) explain
    b) is f differentiable at x = (1/2) explain

    2. Relevant equations

    I have made the graph and x is a point at 1/2 but there is a jump. I have no idea how to start this.



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 8, 2009 #2

    Mark44

    Staff: Mentor

    Drawing a graph is a good start. Can you describe what the graph looks like?
     
  4. Mar 8, 2009 #3
    the line from the left approachs (1/2) at +1 and restarts at (1/2) at -1. There is a single point at (0,1/2)
     
  5. Mar 8, 2009 #4
    sorry it continues to the right at -1
     
  6. Mar 8, 2009 #5

    Mark44

    Staff: Mentor

    No, none of this is right. The graph of this function is in three parts--two horizontal lines and a single point.

    What is f(-1)? f(0)? f(1/2)? f(1)? f(2)? Plotting these points should give you an idea of what the graph of the function looks like.
     
  7. Mar 8, 2009 #6
    okay, I have fixed the graph. I have a line going from (1/2, 1) (1,1)(2,1) ect and a line going from (1/2,-1)(0,-1)(1,-1) ect, and a point at (1/2,0).

    How do i determine if this is continous at 1/2?
     
  8. Mar 9, 2009 #7

    Mark44

    Staff: Mentor

    One quibble. The line doesn't contain the point (1/2, -1). Does the graph look continuous at x = 1/2?
     
  9. Mar 9, 2009 #8
    I would say that the graph is not continuous at x=1/2 as that point does not connect to any other point on the graph.
     
  10. Mar 9, 2009 #9

    Mark44

    Staff: Mentor

    Correct.
     
  11. Mar 9, 2009 #10
    would it then be correct to say that x=1/2 is not differentiable as it is not continuous at that point?
     
  12. Mar 9, 2009 #11
    based on the fact that well every function is not differentiable, very function that is differentiable is continous. Or am I misunderstanding the concept?
     
  13. Mar 9, 2009 #12

    Mark44

    Staff: Mentor

    No. It doesn't make any sense to talk about a point or an x value being differentiable. You can say, though, that a function is continuous or differentiable at a point or at some x value.
     
  14. Mar 9, 2009 #13

    Mark44

    Staff: Mentor

    That is correct, so you are not misunderstanding the concept. I have made a couple of edits to what you wrote:
     
  15. Mar 9, 2009 #14
    B) is f differentiable at x = (1/2) explain

    So I could say that function is not differentiable at x=1/2 as the function is not not continuous?

    Sorry for being a little slow i'm just trying to wrap my head around the concept. Thanks
     
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