  Since Benjamini and Hochberg introduced false discovery rate (FDR) in their
seminal paper, this has become a very popular approach to the multiple
comparisons problem. An increasingly popular topic within functional data
analysis is local inference, i.e., the continuous statistical testing of a null
hypothesis along the domain. The principal issue in this topic is the infinite
amount of tested hypotheses, which can be seen as an extreme case of the
multiple comparisons problem. In this paper we define and discuss the notion of
false discovery rate in a very general functional data setting. Moreover, a
continuous version of the Benjamini-Hochberg procedure is introduced along with
a definition of adjusted p-value function. Some general conditions are stated,
under which the functional Benjamini-Hochberg procedure provides control of the
functional FDR. Two different simulation studies are presented; the first study
has a one-dimensional domain and a comparison with another state of the art
method, and the second study has a planar two-dimensional domain. Finally, the
proposed method is applied to satellite measurements of Earth temperature. In
detail, we aim at identifying the regions of the planet where temperature has
significantly increased in the last decades. After adjustment, large areas are
still significant.
