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\title{Lower Semicontinuous Functions}
\author{By Bogdan Grechuk}
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\begin{abstract}
  We define the notions of lower and upper semicontinuity for functions from a
  metric space to the extended real line. We prove that a function is both lower
  and upper semicontinuous if and only if it is continuous. We also give several
  equivalent characterizations of lower semicontinuity. In particular, we prove
  that a function is lower semicontinuous if and only if its epigraph is a
  closed set. Also, we introduce the notion of the lower semicontinuous hull of
  an arbitrary function and prove its basic properties.
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