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We can now answer the fundamental question 1 on page

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Unformatted text preview: ous manner. And each instant on the TIME AXIS corresponds to a definite instant real number in R. Likewise, when we say x2 TENDS TO x1 on the x-axis, denoted by x2 → x1 , we let point x2 go point , point, point , . . . all the way to point x1 in a continuous point point point point continuous manner. And each point on the x-axis corresponds to a definite real number in R. point From the Algebra point of view R is a COMPLETE set of real numbers. From the Geometry point of view R is a CONTINUOUS line of POINTS. And from the Analysis point of view R is a CONTINUOUS set of INSTANTS. We can now answer the fundamental Question 1 on page 3. In taking the Limit as t2 → t1 w e go from the inter v al [ t1 , t2 ] a nd reach the instant t1. In general, if t is any instant by taking instant, the Limit as δ t → 0 w e can go from the inter v al [t, t+ δ t] and reach the instant t. What happens at an instant is instantaneous. instant instantaneous In English, instantaneous = o ccuring or completed in an instant . i nstantaneous i nstant 20 7. S I N G L E VA L U E D F U N C T I O N S In the next few chapters we apply the concepts TENDS TO, LIMIT and CONTINUOUS INFINITESIMALS to look at the first two properties of WELL-BEHAVED functions, i.e. SINGLE VALUED and CONTINUOUS. To know whether a function is CONTINUOUS or not, we need to know : 1. The LIMIT of a function. 2. The VALUE of a function. With these two concepts in place we define CONTINUOUS functions. A function is said to be SINGLE VALUED* if at every point or instant on the real line where the function is defined it has one and only one value. f(x) f(x) δx (0, 0) a δy a+δ x x Moreover, for an infinitesimal CHANGE δ x at the instant a (where the function is defined) there is exactly one corresponding CHANGE δ y in the value of the function. * Note: In Algebra the concept of a function as developed from a relation (set of ordered pairs) and elation mapping is single valued by definition. In Analysis the concept is not so strict. For calculation purposes we mapping have to enforce this by restricting the range of the function. 21 In the figure below at x = a there are two values of f(x). So, for an infinitesimal CHANGE δ x at...
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