Let's say we have function with empty domain $f:\ \varnothing \rightarrow Y$.

Is this theoretically correct to say that the derivative of this function is another function with empty domain e.g. $g:\ \varnothing \rightarrow Y$?

(I know it's pointless to calculate a derivative of an empty function, but I want to know whether it is formally wrong to say so and if it contradicts with the definition of the deriviative)

  • 2
    $\begingroup$ The derivative isn't even defined in a general set theoretic context. $\endgroup$ – lulu Apr 14 '20 at 17:12
  • $\begingroup$ @lulu sorry I added that tag by mistake. I am asking in the context of (real) analysis. $\endgroup$ – MartinYakuza Apr 14 '20 at 17:20
  • $\begingroup$ Are you assuming $Y$ to be a subset of the real numbers? $\endgroup$ – Servaes Apr 14 '20 at 17:28
  • $\begingroup$ @Servaes yes, but does this really matter? The set of values will be empty anyway. $\endgroup$ – MartinYakuza Apr 14 '20 at 18:08
  • $\begingroup$ You're right that the derivative has empty domain. But "another" can be confusing, since all functions with domain $\varnothing$ are equal, i.e., they're the same set of ordered pairs. $\endgroup$ – Andreas Blass Apr 14 '20 at 19:11

From definition of a derivative we get that the function must be defined at the point under consideration. The defining equation is $$\ f'(x)= \lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$ for sufficiently small h. So here $\ f(x)$ $ \left[ also \ f(x+h)\right]$ has to be defined at some point in the domain. If the domain doesn't contain any point, then this defining equation doesn't make any sense as the functon $\ f(x)$ is not defined at any point. So there is no point in even considering the derivative where the main function is itself not defined.

  • $\begingroup$ I know that this is "pointless", but would it be wrong to say so? That the derivative is an empty function. $\endgroup$ – MartinYakuza Apr 14 '20 at 21:43
  • $\begingroup$ Obviously wrong.. because where the function itself doesn't exist, how can you differentiate that function? $\endgroup$ – Manjoy Das Apr 15 '20 at 6:29
  • $\begingroup$ it exist, it's just empty. $\endgroup$ – MartinYakuza Apr 15 '20 at 10:42
  • $\begingroup$ @MartinYakuza what is an empty function? A set can be empty. How can a function be empty? $\endgroup$ – Manjoy Das Apr 15 '20 at 10:43
  • $\begingroup$ function is a set $\endgroup$ – MartinYakuza Apr 15 '20 at 13:45

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