I have two formula;

With 'a' being a constant, 1>a>0.

I was looking for the equivalent continuous function, that is, functions that would be exactly the same as f(x) and g(x) but would be continuous on [0, +inf

I know that these functions;

...work fine, I'm pretty they're exactly what I'm looking for, but I have no idea how to get (with some rigor) from the discrete form to the continuous form