But what justification do you have for thinking that equation holds
I don’t have one: this is simply filling in the the pattern. You can say “No, for any r>1, there is no answer,” (just like you could reasonably say things like ”−1 has no square root”) but if you decide to extend the domain of the function C, this is not just a possible value, it is the value. (The proof of this claim requires actual rigor, but if you’re willing to go along with pattern matching, you get the right answer.)
I don’t have one: this is simply filling in the the pattern. You can say “No, for any r>1, there is no answer,” (just like you could reasonably say things like ”−1 has no square root”) but if you decide to extend the domain of the function C, this is not just a possible value, it is the value. (The proof of this claim requires actual rigor, but if you’re willing to go along with pattern matching, you get the right answer.)