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

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bert

Posts: 295
Registered: 1/4/06
Re: question on definitions
Posted: Aug 4, 2013 4:16 PM
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On Sunday, 4 August 2013 19:54:49 UTC+1, G Patel wrote:
> 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.
> is this equivalent to? :
> 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?


I don't think so. In your example, if f is NOT a function so defined,
the property of it being continuous or discontinuous may be meaningless.

Consider: if N is an integer, it is said to be prime if ... (usual
definition). Well, when N is not an integer, it is meaningless to
discuss whether it is prime or not prime; it is neither.
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