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Monotonic function

  • #1
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Homework Statement


[tex]y=x^3[/tex]

The Attempt at a Solution


I know that function is increasing when [tex]f'(x) > 0[/tex] but in [tex]x=0[/tex] there is [tex]f'(x) = 0[/tex], so is function increasing there or not? From definition I know that its increasing there, but how can I connect this with theorem that function is increasing when [tex]f'(x) > 0[/tex]?
 

Answers and Replies

  • #2
vela
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What precisely is your definition of an increasing function?
 
  • #3
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[tex]\forall_{x_1,x_2\in X} \left( x_1 < x_2 \Rightarrow f(x_1)<f(x_2) \right)[/tex]
 
  • #4
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If a function is non-decreasing (weakly increasing), and [tex]f(x_1)=f(x_2)[/tex], [tex]x_1\neq x_2[/tex], then f is constant on [tex][x_1, x_2][/tex], so a single zero of a derivative cannot spoil injectivity.
 
  • #5
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aha, so if derivative is positive and 0 only for countable set of points the function will be increasing?
 
  • #6
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Yes. It can probably be strenghtened a little bit, but that should be enough for all practical purpose.
 
  • #7
HallsofIvy
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[tex]\forall_{x_1,x_2\in X} \left( x_1 < x_2 \Rightarrow f(x_1)<f(x_2) \right)[/tex]
With that definition, it makes no sense to talk about a function being "increasing" at a point. It is easy to prove that [itex]y= x^3[/itex] is increasing on any interval.
 
  • #8
114
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thanks for answers. Is any definition which can define increasing in a point?
 
  • #9
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stupid question, sorry, nvm
 

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