  The class $\mathbf{No}$ of surreal numbers, which John Conway discovered
while studying combinatorial games, possesses a rich numerical structure and
shares many arithmetic and algebraic properties with the real numbers. Some
work has also been done to develop analysis on $\mathbf{No}$. In this paper, we
extend this work with a treatment of functions, limits, derivatives, power
series, and integrals.
  We propose surreal definitions of the arctangent and logarithm functions
using truncations of Maclaurin series. Using a new representation of surreals,
we present a formula for the limit of a sequence, and we use this formula to
provide a complete characterization of convergent sequences and to evaluate
certain series and infinite Riemann sums via extrapolation. A similar formula
allows us to evaluates limits (and hence derivatives) of functions.
  By defining a new topology on $\mathbf{No}$, we obtain the Intermediate Value
Theorem even though $\mathbf{No}$ is not Cauchy complete, and we prove that the
Fundamental Theorem of Calculus would hold for surreals if a consistent
definition of integration exists. Extending our study to defining other
analytic functions, evaluating power series in generality, finding a consistent
definition of integration, proving Stokes' Theorem to generalize surreal
integration, and studying differential equations remains open.
