Jo, I disagree about what you are saying concerning continuity. There is 'continuous' and there is 'continuous by continuation' (again, I'm not certain that this is the english term). In practice by far only the second definition is used.
In the strictest sense 'continuous' is exactly what you described. But you wont find anybody that says that sinc(x)=sin(x)/x is discontinuous. It's because there is a well defined *unique* way to assign a value to the 'false' discontinuity at zero. (And the plot is quite nice at that 'discontinuity' ;) The unique procedure is as follows: Consider f(x) that is not defined at x_0. If the left and right limits of f at x_0 exist and are equal you can define f(x_0) as that limit. Consider the example of sinc(x) - should we really treat it differently than sin(x)/x. As I said, I agree with you about the strictest definition, but it's just not the one that is used by people. And there is a good reason: the fact that there is a unique and well defined procedure to remove the discontinuity means that this discontinuity is just an artefact. And practice shows that it does not really matter. On 11 January 2012 21:44, Joachim Durchholz <[email protected]> wrote: > Am 11.01.2012 11:32, schrieb Alexey U. Gudchenko: > >> 11.01.2012 12:51, Joachim Durchholz пишет: >> >>> x²/x has a discontinuity for x=0, and x does not, hence x²/x is not the >>> same as x. >>> >> >> No, no, it is continuous because the limit when x-->0 exists (equals 0), >> > > The limit exists, but that's just half of the definition of "continuous". > > > > and the same as a value of function at this point, 0**2/0 (which by > >> definition is equal 0). >> > > f(x) = x²/x has no definition for x=0. It involves division by zero. > > If you go from functions to relations, then for a, b != 0, we have > - a/b is a one-element set > - a/0 is the empty set since no r satisfies 0*r = a > - 0/0 is the the set of all values since all r satisfy 0*r = 0 > So you can assign an arbitrary value to the result of 0/0 and it will be > "correct", but you don't know whether the value you assigned is "more" or > "less" correct than any other. > For example, what should (x²/x)/(x²/x) be for x=0? > If you do the x-->0 limit first, you'll get 1. > If you stick with the basic substitutability rules of math, you get > (x²/x)/(x²/x) = (0)/(0) = 0/0 = 0. > Hilarity ensues. > > Oh, and I bet different people will have different assumptions about which > rule should take priority. > And if their assumptions differ from those that simplify() applies, they > will come here and ask what's wrong. > > Regards, > Jo > > > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To post to this group, send email to [email protected]. > To unsubscribe from this group, send email to sympy+unsubscribe@** > googlegroups.com <sympy%[email protected]>. > For more options, visit this group at http://groups.google.com/** > group/sympy?hl=en <http://groups.google.com/group/sympy?hl=en>. > > -- You received this message because you are subscribed to the Google Groups "sympy" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. For more options, visit this group at http://groups.google.com/group/sympy?hl=en.
