Saturday, 27 October 2018

Searching function starting exactly constant and approaching another constant

for the default of an R API parameter i seek a function that has the property of yielding a good guess.




  • I want the function to be defined for Z+ (But no reason not to define it for R+, i guess)

  • I want it to be smooth

  • It should satisfy both




    f(xsmall)=1xsmall(0,k]
    lim




edit to be clear: k and l should be constants appearing in the function definition.



E.g. with k = 1000 and l = 0, the f(x) = 1 \forall x \in (0, 1000]. Then it should gradually decline and approach 0.



To simplify, the function can be written as:




\mathrm f(x) = \begin{cases} 1 & \text{if } x \le k \\ [\dots] & \text{otherwise} \end{cases}

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