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)=1∀xsmall∈(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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