How can I prove this from Roger Penrose's Road to Reality?

[Etienne]

Hi, I've been reading Roger Penrose's Road to Reality during my free time, and I am trying to do all the proofs I possibly can, although I am quickly reaching my limit.

Would somebody help me prove this?

h(x)=0 if x ≤ 0
h(x) = e^-1/x if x > 0

Thank you in advance :)

 

[BvU]

Hi,

Read what you wrote and then maybe understand that there is no question in your post. Prove what ?

 

[mfb]

This is an incomplete definition of h(x). You cannot prove a definition.
You can prove that h is continuous by looking at a specific limit.

 

[Etienne]

I'm so sorry, I see I only wrote the definition of the function.

It is said that the function h(x) is of class C^∞ in the domain ℝ. What I understand is that it must have an infinite number of derivatives, and that at the slope must be 0 at the origin.

However, I haven't been able to prove this and apparently neither could a friend. It is probably much easier than I think, but I haven't been able to do it.

Could I approach this with a Taylor series? Logarithms? I apologize for the anterior and thanks again.

 

[Demystifier]

Prove that it is $C^1$ and then use induction to prove that it is $C^n$ for all $n\geq 1$.