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pause to give it a precise mathematical definition. In a string of beads, the string is
continuous and the beads are discrete. Yet, from ancient times philosopherdiscr
mathematicians were aware of the concept “continuous” and tried to define it.
In his Physik, Aristotle (384-322 BC), Greek philosopher and student of Plato and
tutor to Alexander the great, explained “continuous” as: ‘ I say that something is
continuous whenever the two extremities of their contiguous parts coincide, and as
the name itself implies, they are kept together. ’
Gottfried Wilhelm von LEIBNIZ (1646-1716), co-inventor of Calculus and librarian
and historian under Duke Johann Friedrich of Hanover, defined “continuous” as:
‘The whole is said to be continuous, when any two component parts thereof (or more
precisely any two parts which together make up the whole) have something in common,
... at the very least a common boundary.’
It is interesting that Sir Isaac Newton did not have much to say on “continuous” .
More recently R. DEDEKIND (1872) defined “continuous” as: ‘If the points of a
line are divided into two classes, in such a way that each point of the first class lies to
the left of every point of the second class, then there exists one and only one point
of division which produces this particular sub-division into two classes, this cutting of
the line into two parts.’
We saw the connection between the set of Real numbers being complete and the
Real line being continuous. The absence of a single point causes a cut or break or
discontinuity in the Real line.
In Calculus we deal with continuous operands or functions. We now carry forward
the concept of the Real line being continuous to functions over the Real line. We
need to know the behaviour or value of a function at each and every point or
instant. Also, we need...
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- Fall '09