The function g
-times differentiable, the
-th derivative is continuous, and g
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
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
once differentiable, the derivative is continuous, but not differentiable
again. Why ? Then try n = 2
. Then, maybe, try an induction argument
to prove the general case.