Re: ** says: Definition: sum{i in N} i = 0

In article <1184316489.877675.232710@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
WM <mueckenh@xxxxxxxxxxxxxxxxx> wrote:

On 12 Jul., 01:52, Virgil <vir...@xxxxxxxxxxx> wrote:
In article <1afb1$46948483$82a1e228$10...@xxxxxxxxxxxxxxxx>,
Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:





Virgil wrote:

In article <1184096471.659205.71...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:

The derivative of |x| is not

continuous but -1 for negative x and +1 for positive x. In contrast
sinx/x is continuous. Your example fails.

The expression sin(x)/x is not even defined for x = 0, so that WM is
claiming that it has a value when it does not have a value.


It IS defined for x = 0 and its value IS always 1 at x = 0 .


There are two similar functions, one of which is not even defined at x =
0 and another which is continuous, and even differentiable, there.-

And there are uncountably many others with f(x) = r in R. But only one
of these functions exists in useful mathematics. And f(x) for this one
can be determined by l'Hospital.

On the contrary, the "other" one (undefined at 0) is quite useful in the
teaching of mathematics.

And one of the things it teaches is that the expression sin(x)/x is not
defined at x = 0.

What may transpire in physics usage is occasionally invalid in
mathematics, as in this instance.
.

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