/sci/ - Science & Math

>>11225443
Suppose I have a function f, but I want to smoothly "skip over" one of the integer values of x. For example, if I have the function f(x) = 2x, but I want to skip the point (3, 6), so that the actual value of the function at x=3 is 8. So make a new function h(x) which takes on the following values:
h(1) = 2
h(2) = 4
h(3) = 8
h(4) = 10
h(5) = 12
etc.

In pic related, I am almost there. I make a sigmoid function s(x) centered at x=2.5. And I make a function g which is just f shifted over to the left one. And I use the sigmoid to transition from one to the other. But this is not an exact solution- it only asymptotically approaches the exact values I want. I would much prefer an exact solution to this puzzle (which is still smooth). I think all I need is to replace the sigmoid with a smooth function that is exactly 0 at x ≤ 2 and exactly 1 at x ≥ 3.

Can anybody help me out here?