In Reply to: indefinitely differentiable posted by Brian on February 23, 2004 at 01:10:12:
>Exercise 4 of Chapter 5 asks us to show that f is indefinitely differentiable on R. I'm not familiar with this term "indefinitely differentiable". Does this mean to show f is differentiable for all x in R?
No; for that we would just say that f is "differentiable".
"Indefinitely differentiable" (also called "infinitely differentiable") means that not only is f differentiable for all x,
but that its derivative f' is also differentiable (i.e. f is
twice differentiable), that the second derivative f'' is also
differentiable, and so forth.
For instance, the function sin(x) is indefinitely differentiable
because you can differentiate it as many times as you want
(i.e. you can keep differentiating it indefinitely). The function
|x|^3 is twice differentiable but not three times differentiable
(and hence not indefinitely differentiable).
Terry
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