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Homework Help: Continuous Function?

  1. Jan 10, 2010 #1
    EDIT: just realised i might've been really stupid;

    very simple question which will answer my stupidly long question;

    is f(x) = 1 continuous?]

    The reason I ask is that my book says;

    [tex] f(x,y) \in C^{N}[/tex] in R [tex]\Leftrightarrow \frac{\partial ^{n} f}{\partial x^n} , \frac{\partial ^{n} f}{\partial x^{n-1}\partial y}, etc \in C[/tex] in R.

    however you can imagine for f (x, y) = x which IS continuous,

    f1(x, y) = 1 , would this be considered continuous, and if not, isn't that at ends with the definition above?
     
    Last edited: Jan 10, 2010
  2. jcsd
  3. Jan 10, 2010 #2

    Dick

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    Of course it's continuous. lim (x1,y1)->(x,y) f(x1,y1)=1. f(x,y)=1. It's easy using any definition of continuity. Look one up!
     
  4. Jan 10, 2010 #3
    Yes, I thought as much, but why then would;

    f(x) = x (for x greater than/equal to 0) / 0 (for x less than 0)

    Why then for this function would

    [tex]
    f(x) \notin C^{1}
    [/tex]

    ?

    Would it be because there is a jump, a "discontinuity" at 0, where f(x) goes from 1 to 0 instantly?
     
  5. Jan 10, 2010 #4

    Dick

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    Sure. f(x) is continuous. f'(x) is not continuous (I'm assuming you meant to say f'(x)). So it's not in C^1.
     
  6. Jan 10, 2010 #5
    Hmmm so it's not continuous?

    But then sureley it is at ends with the definition posted earlier; namingly

    [tex]
    f(x,y) \in C^{N}
    [/tex] [tex]
    \Leftrightarrow \frac{\partial ^{n} f}{\partial x^n} , \frac{\partial ^{n} f}{\partial x^{n-1}\partial y}, etc \in C
    [/tex]

    because df/dx is not continuous, so it is implied that f(x) is not continuous, or is there a flaw in my reasoning? :confused:

    edit: and yeah sorry i meant f'(x)
     
  7. Jan 10, 2010 #6
    sorry I just realised i'm being a major douche...i'm stupid, nevermind
     
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