alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

In calculus we deal with continuous operands or

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Unformatted text preview: ldom continuous pause to give it a precise mathematical definition. In a string of beads, the string is continuous discrete continuous and the beads are discrete. Yet, from ancient times philosopherdiscr mathematicians were aware of the concept “continuous” and tried to define it. “continuous” In his Physik, Aristotle (384-322 BC), Greek philosopher and student of Plato and Physik tutor to Alexander the great, explained “continuous” as: ‘ I say that something is “continuous” 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 “continuous” and historian under Duke Johann Friedrich of Hanover, defined “continuous” as: “contin ‘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” . “continuous” More recently R. DEDEKIND (1872) defined “continuous” as: ‘If the points of a “continuous” 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 complete Real line being continuous. The absence of a single point causes a cut or break or continuous discontinuity in the Real line. In Calculus we deal with continuous operands or functions. We now carry forward continuous contin the concept of the Real line being continuous to functions over the Real line. We continuous contin need to know the behaviour or value of a function at each and every point or point instant. Also, we need...
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