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Here is a short question: finding a function that 1) $f(0)=1$ and 2) $f(x)=0$ for all $x>0$.

I have two functions in mind:

  1. Something like Dirac delta: $\delta_0(x)$.
  2. Something like $f(x)=1-\text{sign}(x)$.

But they are (arguably) not elementary. A function that is simpler and more elementary will be appreciated!

dodo
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  • Why not just use the function you defined? It's not the dirac delta (as that "function" is certainly not defined at $0$). You can define it any way you like for $x<0$ (or leave it undefined if you prefer). – lulu May 08 '22 at 18:09
  • What functions do you consider elementary? – paperskilltrees May 08 '22 at 18:15
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    @paperskilltrees Any simple functions that are learned or easily understood by high-schoolers. I think it might be impossible as any operations learned in high school are continuous, and this function is not continuous. – dodo May 08 '22 at 18:17
  • @paperskilltrees Or maybe we can use the definition here: https://math.stackexchange.com/questions/118113/what-makes-elementary-functions-elementary – dodo May 08 '22 at 18:19
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    You are absolutely correct in that observation! Any sum, product, superposition etc of continuous functions is continuous. Division by zero won't give you the kind of discontinuity you are looking for. – paperskilltrees May 08 '22 at 18:24

1 Answers1

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You can take the example

$$f(x)=\lfloor e^{-x} \rfloor$$

So

$$f(0)=\lfloor 1\rfloor=1$$ and for $ x>0$

$$0<e^{-x}<1\implies f(x)=0$$

hamam_Abdallah
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