Unformatted text preview: ous manner. And each instant on the TIME AXIS corresponds to a definite
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
manner. And each point on the x-axis corresponds to a definite real number in R.
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
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.
In English, instantaneous = o ccuring or completed in an instant .
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.
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
* Note: In Algebra the concept of a function as developed from a relation (set of ordered pairs) and
mapping is single valued by definition. In Analysis the concept is not so strict. For calculation purposes we
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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- Fall '09
- Limit, Δx