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Re: question on definitions
Posted:
Aug 4, 2013 9:48 PM


On Sun, 4 Aug 2013, G Patel wrote:
> Some definitions I read are like follows: > definition of continuity of a function at x:
f is continuous at x when for all open V nhood f(x), there's some open U nhood x with f(U) subset V.
> Let f be a function defined on an open interval of x.
> Now/then, f is said to be continuous at x if/iff some predicate P(f,x) is > true.
Too vague for an intelligent response. > f is said to be continuous at x if/iff f is a function defined on an open > interval of x and some predicate P(f,x) is true.
> That is, when the first statement is made ahead of the definition proper, > how do we interpret it? If some point does not meet the initial statement, > are we allowed to call the function "discontinuous" at that point?
That f be defined about an interval of x isn't needed except perhaps for those doing analysis. For example f:{ 0, 1/n  n in N } > R, 1/n > 1/2n if n in N, 0 > 1 is continuous at every point of the domain of f except at 0. If instead 0 > 0, then f is continuous.



