[FOM] Query on real functions

[JoeShipman@aol.com]

Can anyone identify a real-valued function f continuous on [1, infinity) with the following two properties:

1) f eventually dominates any function in the sequence e^x, e^(e^x), e^(e^(e^x)), ....

2) f is defined in some other way than by defining it first at all integers and then interpolating

Defining it first for some dense set, like dyadic rationals, and then interpolating, is of course acceptable.  That is the standard way to extend the power function, defined by induction, to a continuous function, by identifying the square root with the "one-halfth" power, etc.

Unfortunately, a nice way to do this for "tower" rather than "power" is not apparent: 2T1=2, 2T2=4, 2T3=16, 2T4=65536, but what could 2T(3.5) possibly be?


-- JS