On Sunday, August 4, 2013 2:54:49 PM UTC-4, G Patel wrote: > Some definitions I read are like follows: definition of continuity of a function at x: ------------------------------------------------ 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?
There are various definitions of continuity depending upon the particular pair of sets upon which the function is based and on some other things as well. You are asking a good question, but I suspect that you are lacking a certain amount of background in mathematics more advanced than calculus. This background makes everything a lot clearer, but, certainly when I was taking calculus I was presented with the same type of definitions that I think you are getting.
If you are dealing with the usual way of looking at real numbers, continuity at a point requires the function to be defined on an open set containing the point. However, there are also definitions for left continuity and right continuity, each of which requires an open set on one side of the point or the other. There is a definition of the continuity of a function on an open set rather than at a specific point. It turns out to be equivalent to the function being continuous using your definition, but at every point in the open set. This actually turns out to be the definition that has been most useful for generalizing the definition of continuity to functions other than ones that take real numbers to real numbers.
I will be happy to tell you more, but I don't know what you know nor do i have adequate information about what yhou are really interested in to tell you much more.