Question about a definition

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The definition
Let a, b be any two real numbers
Let c > 0
We define a function f to be funny iff
For all x, y belonging to [a,b], |f (x) - f (y)| ≤ c |x - y|

Question
Let a < b (arbitrarily)
Let c > 0
Assume function g is funny on [a, b]
Let x, y ∈ [a, b]
Therefore, |g (x) - g (y)| ≤ c |x - y|
= > |g (x) - g (y)| / |x-y| ≤ c

I'm confused at this part because c is arbitrary. Does the original definition mean that a function is funny for all c > 0 or for some c > 0? And if it is for some c > 0, what is the statement saying about the function? Is it just stating that the secant of the function is less than equal to some positive number? Can there be a function that is not funny if this is the case?
 

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  • #2
Math_QED
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The definition
Let a, b be any two real numbers
Let c > 0
We define a function f to be funny iff
For all x, y belonging to [a,b], |f (x) - f (y)| ≤ c |x - y|

Question
Let a < b (arbitrarily)
Let c > 0
Assume function g is funny on [a, b]
Let x, y ∈ [a, b]
Therefore, |g (x) - g (y)| ≤ c |x - y|
= > |g (x) - g (y)| / |x-y| ≤ c

I'm confused at this part because c is arbitrary. Does the original definition mean that a function is funny for all c > 0 or for some c > 0? And if it is for some c > 0, what is the statement saying about the function? Is it just stating that the secant of the function is less than equal to some positive number? Can there be a function that is not funny if this is the case?
For a fixed c. This property is also called Lipschitz-continuity.

Only constant functions are funny for all c (exercise).
 
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  • #3
jbriggs444
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The definition
Let a, b be any two real numbers
Let c > 0
We define a function f to be funny iff
For all x, y belonging to [a,b], |f (x) - f (y)| ≤ c |x - y|
As I interpret the above definition, "funny" is a boolean-valued function with four parameters: a, b, c and f.
Question
Let a < b (arbitrarily)
Let c > 0
Assume function g is funny on [a, b]
The usage here suggests that "funny" is to be viewed as family of functions. A member of the family is selected based on a fixed level of funniness (c) and a fixed interval over which it is applied ([a,b]). Then that family member decides which functions are funny (at level c on interval [a,b]) and which are not.
 
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  • #4
HallsofIvy
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The fact that the problem uses the word "funny" would make me leery or the whole thing! Yes, that definition should say "there exist c such that ---".
 
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The fact that the problem uses the word "funny" would make me leery or the whole thing! Yes, that definition should say "there exist c such that ---".
Although I think this is purposeful since the actual question wanted to prove integrability of f. Changing the quantifier to a for all in the Lipschitz definition I guess is just another way to say that a function is constant.

Nevermind, this is correct but the question I had before is not saying that.
 
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