  In his monograph On Numbers and Games, J. H. Conway introduced a real-closed
field No of surreal numbers containing the reals and the ordinals, as well as a
vast array of less familiar numbers. A longstanding aim has been to develop
analysis on No as a powerful extension of ordinary analysis on the reals. This
entails finding a natural way of extending important functions f from the reals
to the reals to functions f* from the surreals to the surreals, and naturally
defining integration on the f*. The usual square root, log, and exp were
naturally extended to No by Bach, Conway, Kruskal, and Norton, retaining their
usual properties. Later Norton also proposed a treatment of integration, but
Kruskal discovered it has flaws. In his recent survey [2, p. 438], Siegel
characterizes the question of the existence of a reasonable definition of
surreal integration as "perhaps the most important open problem in the theory
of surreal numbers." This paper addresses this and related unresolved issues
with positive and negative results. In the positive direction, we show that
semi-algebraic, semi-analytic, analytic, meromorphic, or more generally
\'Ecalle-Borel transseriable functions extend naturally to No, and an integral
with good properties exists on them. In the negative direction, we show there
is a fundamental set-theoretic obstruction to naturally extending many larger
families of functions.
