On 10/20/2013 4:07 PM, quasi wrote: > Hetware wrote: >> >> So Quasi was correct. > > Of course! > > (just kidding) > >> I can define a function g(x) to be continuous, > > I don't think I said that.
Someone agreed with me that a function can be defined as continuous.
> Let's try a similar example with a change of wording: > > Let h be a continuous function such that > > h(x) = f(x) for x != 3 > > Does the above qualify as a "definition" of h? > > For the above to be regarded as a definition, it must be the > case that there is one and only one function h which satisfies > the above conditions. But you would first have to _prove_ that > claim. Only then can you claim to have "defined" h. > > In other words, imposing conditions on h is not the same as > defining h.
What I intend by defining a function to be continuous is to remove all removable discontinuities resulting from the (other) rules given. We are always obliged to determine that the definition proposed does indeed satisfy the properties of a function as well as consistency. So if we proposed a definition which asserted continuity, and other rules of the function resulted in an irremovable discontinuity, the proposed function would be invalid.
> Now consider a sample test problem ... > > Suppose k(x) is continuous on R and such that > > k(x) = (x^2 - 9)/(x - 3) > > for x != 3. Must k(3) = 6? > > The answer is yes. > > But the question didn't _define_ k(x). Rather, it specified > some conditions on k(x) and then asked a question regarding > such a function k. > > quasi >
I could include a rule in the definition of the function which says wherever there is a removable discontinuity, remove it. Can you give me an example where including such a rule would not result in a set of ordered pairs satisfying the requirements of a continuous function? That is assuming a function defined identically with the exception of that rule results in a function with no more than a finite number of removable discontinuities and not irremovable discontinuities?
For example, let f(x) = (x^2 - 9)/(x - 3) where (x^2 - 9)/(x - 3) is defined, else f(x) = limit[(s^2 - 9)/(s - 3), s->x].