Is there an infinitely differentiable function $f$ such that

  • $f(x)=0 \iff x=0$,
  • $f$ is infinitely differentiable
  • No derivative of $f$ is ever equal to zero.
  • 3
    $\begingroup$ Think about what you can do with the exponential function. $\endgroup$
    – Git Gud
    Oct 10, 2014 at 18:23

1 Answer 1


Consider the function $f:\Bbb R \to \Bbb R$ s.t $f(x)=e^x-1$.

  • $\begingroup$ Precisely what I thought about when I saw the question. (+1) $\endgroup$
    – robjohn
    Oct 10, 2014 at 19:03

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