C(n) Function

The function g is n-times differentiable, the n-th derivative is continuous, and g is not (n+1)-times differentiable.
Back Back

On the right are the graphs of this root function for n = 0, 1 and 2. Can you decide which graph goes with what n ? Click on Options and choose different powers of x to see how the function gets smoother and smoother at 0, which is necessary if it wants to be more and more times differentiable.

Your browser can not handle Java Applets ...

Proof:

This proof is left as an exercise. As a hint, start with n = 1. The example, if correct, says that for n = 1 the function g is once differentiable, the derivative is continuous, but differentiable again. Why ? Then try n = 2. Then, maybe, try an induction argument to prove the general case.

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 28, 2007