Does the logistic function uniquely satisfy these three conditions?Givenr(t)=f(t)1−F(t) {Eq. 1}wheref(t)=dFdt {Eq. 2}and the conditions:limt→∞r(t)=1 {Eq. 3}limt→∞F(t)=1 {Eq. 4}limt→∞f(t)=1−F(t) {Eq. 5}I can think of just one function F satisfying these three conditions--the logistic function:F(t)=11+e−t {Eq. 6}(which can also be expressed F(t)=r(t))Is this is the only function satisfying these conditions? If so, is there a way to prove it?

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Boehm98wy · Accepted Answer

r(t)=F′(t)1−F(t)r(t),dt=dF1−F∫r(t),dt=−log⁡(1−F(t))+C1−exp⁡(−∫r(t),dt+C)=F(t).So you can put as many things in the role of F as will fit in the role of r, i.e. things satisfying your Eq. 3.

Does the logistic function uniquely satisfy these three

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