Average value of a function question

  • Thread starter drpizza
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  • #26
Gib Z
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matt - ahh I didn't mean the pencil thing as a definition...just a novice way to maybe think of it...I know its wrong now, please stop punishing me :P

Q is the natural numbers right? I don't understand when you say it is a function from Q to Q...But I get the idea that only natural numbers are allowed, thanks for that good example.
 
  • #27
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Am I missing something??
Think about functions with their domain and range as the rational numbers. f(x) = x^2, where all of the inputs are fractions, and all the outputs are fractions.

Applying your definition of continuity makes sense in this case, because we can talk about the limit from right and the left (limits make sense in the rational numbers). We say in this case that f is continuous, meaning that f is continuous over its domain. The fact that f is not defined at e or pi does not make it discontinous at these points (remember the domain of f is fractions).

Instead of saying that the rational function is 'discontinuous at x = 2' (has no content by the definition) say that it is 'unbounded in any interval containing 2' (has content and expresses bad behavior of the function).
 
  • #28
matt grime
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You could call f(x) a continuous function, because it's continuous over its entire domain, but it's not continuous on the interval [0,4], because 2 isn't in its domain. Am I missing something??
You're missing the fact that with this definition there is no such thing as a continuous function - there is always something 'not in the domain' of a function, be it 2, i, E_10, R^4, whatever.
 
  • #29
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Okay, I understand what you're saying. Thanks!

But, I'm still having trouble with what I'm missing when an interval is specified, such as [0,4]; wouldn't that be implying over the real numbers? i.e. i is not included?
 
  • #30
matt grime
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But what does that have to do with anything?
 

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