The existence and uniqueness of a maximum point for a continuous real—valued function on a metric space are investigated constructively. In particular, it is shown, in the spirit of reverse mathematics, that a natural unique existence theorem is equivalent to the fan theorem.
"The fan theorem and unique existence of maxima." J. Symbolic Logic 71 (2) 713 - 720, June 2006. https://doi.org/10.2178/jsl/1146620167