  We study the set of continuous functions that admit no spurious local optima
(i.e. local minima that are not global minima) which we term \textit{global
functions}. They satisfy various powerful properties for analyzing nonconvex
and nonsmooth optimization problems. For instance, they satisfy a theorem akin
to the fundamental uniform limit theorem in the analysis regarding continuous
functions. Global functions are also endowed with useful properties regarding
the composition of functions and change of variables. Using these new results,
we show that a class of nonconvex and nonsmooth optimization problems arising
in tensor decomposition applications are global functions. This is the first
result concerning nonconvex methods for nonsmooth objective functions. Our
result provides a theoretical guarantee for the widely-used $\ell_1$ norm to
avoid outliers in nonconvex optimization.
