> “Linear” means continuous<p>No. As in G. Simmons, the two pillars of
the field of 'analysis' in math are
linearity and continuity. The two are
quite different.<p>Linearity usually has to do with numbers
and vectors. The numbers are usually in
the set R of real numbers or the set C of
complex numbers. Commonly the numbers are
called scalars. Then, a function f is
'linear' provided for scalars a and b and
vectors x and y we have<p>f(ax + by) = af(x) + bf(y)<p>The role, utility? It's an enormous,
powerful simplification and, in
particular, is crucial to systems (high
school style) 'systems of linear
equations', linearity, essentially 'super
position', in quantum mechanics, and
linear operators, as in the classic
Dunford and Schwarz, e.g., as in the
results of S. Banach. In calculus,
differentiation and integration are both
linear operators.<p>And, in particular, linearity is crucial
in both linear programming and integer
linear programming. Local linear
approximations are crucial in non-linear
optimization, e.g., the Kuhn-Tucker
conditions and their associated constraint
qualifications.<p>Continuity is also a biggie, e.g., for a
function to be continuous on a compact
set, e.g., [0,1], means it is also
uniformly continuous, bounded, and
achieves its greatest lower bound and
least upper bound. Continuity,
compactness, and uniform continuity are
the standard assumptions that guarantee
that the Riemann integral of freshman
calculus exists.<p>Maybe what was meant was usually when we
mention linear programming we have, say,
find x to solve Ax = b, x >= 0 where x,
where for positive integer n, is n x 1
and, thus, for the set of real numbers R,
in the set R^n. Here the A is a matrix,
say, for some positive integer m, m x n.
Then from the beginnings of the start of
matrix theory, for real a and b and n x 1
y we have<p>A(ax + by) = aAx + bAy<p>which says that A is a linear function
(operator, transformation, etc.). That's
the 'linearity' in linear programming.
And it holds in linear integer programming
where we ask that some or all of the
components of x be integers.