  We characterize structures such as monotonicity, convexity, and modality in
smooth regression curves using persistent homology. Persistent homology is a
key tool in topological data analysis that detects higher dimensional
topological features such as connected components and holes (cycles or loops)
in the data. In other words, persistent homology is a multiscale version of
homology that characterizes sets based on the connected components and holes.
We use super-level sets of functions to extract geometric features via
persistent homology. In particular, we explore structures in regression curves
via the persistent homology of super-level sets of a function, where the
function of interest is - the first derivative of the regression function.
  In the course of this study, we extend an existing procedure of estimating
the persistent homology for the first derivative of a regression function and
establish its consistency. Moreover, as an application of the proposed
methodology, we demonstrate that the persistent homology of the derivative of a
function can reveal hidden structures in the function that are not visible from
the persistent homology of the function itself. In addition, we also illustrate
that the proposed procedure can be used to compare the shapes of two or more
regression curves which is not possible merely from the persistent homology of
the function itself.
