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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?