function whose cube is smooth

Hi,

I want to charectize the function whose cube is smooth from R to R. For example x^1/3 is smooth and olsa any polynomial but how can i charectrize it?

Thanks

Answer 1

[Jan]

The cube of smooth function is smooth.

Answer 2

[Walter Roberson]

Note that the polynomial roots does not necessarily have to be restricted to reals in order to map R->R . For example,

x*(x-i)*(x+i)

is

x*(x^2+1)

which is

x^3 + x

which maps R -> R

One cannot simply say "polynomials" because not every polynomial with complex roots is going to map R -> R . One thus might need to characterize which polynomials with complex roots are suitable.

[Amit Kumar]

Is my concept right ???

f(x)=x is smooth, but f(x)=x1/3 is not. And f(x)=x3 is not a diffeo. from ℝ to ℝ , since its inverse x1/3 is not differentiable at 0.

[Walter Roberson]

I do not recognize the term "diffeo." ?

Answer 3

[Amit Kumar]

i just want to ask what are the functions whose cube is smooth?