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. --