turns ratio

The only functions for which a derivative isn't defined are discontinuous, no? Can a continuous function have a discontinuous derivative?

pile

BAD link!

a better one:

That one works.

Sure, try:

f(x) = 0 when x < 0 = x^2 when x >= 0

Then, f'(x) = 0 when x < 0 = 2x when x >= 0

and f''(x) = 0 when x < 0 = 2 when x >= 0

f(3')(x) is zero everywhere but at x = 0, where it is undefined.

See also this function, for instance:

Each term of the series is differentiable, but the infinite sum is not.

Note that using the rules for derivatives works, term by term, but the definition does not. This is because, using the definition of the derivative, the limit is taken simultaneously over all terms together, whereas if you evaluate each term independently, you're rearranging the series, which changes its convergence.

If you would like an example of a function which is only n-differentiable, while the function and all n derivatives are continuous, this function should be integrable as well (I suspect a proof using Riemann sums should show it will converge). Simply integrate n times to obtain the desired functions.

Tim

--
Deep Friar: a very philosophical monk.
Website: http://webpages.charter.net/dawill/tmoranwms

exist

discontinuous,=20

n-differentiable,=20

should=20

While i did not have continuity in mind, it is a fairly obvious differentiability killer. Integratability is a different pan of sknarr.

?-)

Octave (a free Matlab clone, very handy around the chicken coop, get it from octave.org) has the rats() rational approximation function:

rats(1.0943,5) ans = 58/53

rats(1.0943,10) ans = 4166/3807

rats(pi) ans = 355/113

rats(pi,4) ans = 22/7

and the fourth, fifth and sixth derivatives of position are also used, and called Jounce or Snap, Crackle, Pop

They are used, e.g. in servo control, when doing velocity or torque control loops.