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

On 11 Jul., 12:10, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:
On 10 Jul., 22:44, Virgil <vir...@xxxxxxxxxxx> wrote:



In article <1184096471.659205.71...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,

WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:
On 9 Jul., 21:56, Virgil <vir...@xxxxxxxxxxx> wrote:

Then according to WM, every derivative must always equal 1 at all
points, and f(x) = |x| must have a derivative at x = 0, and lots more.

That's nonsense.

Maybe, but it is WM's nonsense, not mine, to argue that functions must
continuous at points outside their domains.

Continuous functions must be continuous.

No function is continuous at any point outside its domain of definition,
however continuous it may be within that domain.

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.

The function sin(x)/x does not care a damn about your judgement
whether it is defined or not at x = 0. It is defined there since 1696
when l'Hospital wrote his book.

I really need to look up the original and see
if he was really formulating so carelessly.
What I have learned at school under the keyword
l'Hospital is more along following lines:
If f,g are differentiable functions such that the limits
lim_{x->0} f(x) and lim_{x->0) g(x) exist and are 0 and
lim_{x->0} f'(x)/g'(x) exists,
then lim_{x->0} f(x)/g(x) exists
and we have
lim_{x->0} f(x)/g(x) = lim_{x->0} f'(x)/g'(x).
Nothing is said here about f(0)/g(0) having
a single true value or being defined.




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