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Unformatted text preview: TIME
AXIS is a CONTINUOUS set of INSTANTS and that each INSTANT corresponds to a
Real number. We shall then define CONTINUOUS functions. 4 Par t 1 : CONTINUITY 5 Over view
Ov er vie w continuous
To show that the TIME AXIS is a continuous set of instants and that each instant
corresponds to a Real number we shall proceed in three steps.
Step 1: We know from class 8 Geometry that a straight line is a continuous set of
points. We are also familiar with the different kinds of sets of numbers in Algebra:
N (natural numbers), Z (integers), Q (rational numbers), Irrational numbers and
R (real numbers). We present the Analysis concept of the COMPLETENESS property
of the Real numbers.
We thus show the connection between the set R of real numbers in Algebra and a
straight line (a continuous set of points) in Geometry. Each point xi on the number
line corresponds to a Real number and vice versa. We may call this number line the
x-axis. And when we say x1 < x2 for x1, x2 ∈ R , the picture from the Geometry point
of view is :
≡ set R x-axis x1 < x2 x1 x2
Step 2: In this step we learn the concepts: TENDS TO, LIMIT and INFINITESIMAL.
We shall use these concepts to take an Analysis view of the x-axis as a continuous
set of points. We shall then define an instant on the TIME AXIS. From this we shall
show the equivalence between the x-axis as a continuous set of points and the
TIME AXIS as a continuous set of instants.
x1 x2 ANALYSIS
≡ time axis
t1 t2 Each point xi on the x-axis corresponds to an instant ti on the TIME AXIS.
Step 3: Here we relate the TIME AXIS to the set of real numbers R. Now when we
say instant t1 we mean some definite real number t1 ∈ R. And when we say t1 < t2
for t1, t2 in R , the picture from the Analysis point of view is :
set R ANALYSIS
≡ time axis t1 < t2 t1 t2 Since the TIME AXIS is a continuous set of instants, we may say t2 TENDS TO t1 in
a continuous manner. This is denoted by t...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.
- Fall '09