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

  1. Oct 20, 2016 #1
    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 :)
  2. jcsd
  3. Oct 20, 2016 #2


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    Read what you wrote and then maybe understand that there is no question in your post. Prove what ?
  4. Oct 20, 2016 #3


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    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.
  5. Oct 20, 2016 #4
    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. 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.
  6. Oct 21, 2016 #5


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