  Left and right-continuous functions play an important role in Real analysis,
especially in Measure Theory and Integration on the real line and in Stochastic
processes indexed by a continuous real time. Semi-continuous functions are also
of major interest in the same way. This paper aims at presenting a useful
handling of semi-continuous function in parallel with the way left or right
continuous functions are treated. For example, a lower or upper continuous
function shares the property that it is measurable if it fails to be upper or
lower continuous at most at a number of countable points with a left or
right-continuous when it fails to be left or right-continuous at most at a
number of countable points. As a final result, the comparison between the
Riemann and the Lebesgue integrals on compact sets is done in a very
comfortable and comprehensive way. The frame used here is open to more further
sophistication
