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Topic: question on definitions
Replies: 3   Last Post: Aug 4, 2013 9:48 PM

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William Elliot

Posts: 1,448
Registered: 1/8/12
Re: question on definitions
Posted: Aug 4, 2013 9:48 PM
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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.



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