Unformatted text preview: An each such point of discontinuity f(x) has 2 definite values.
discontinuity
3. Over each of the (n + 1) subintervals the continuous summation must be
continuous summation
contin
something finite, say : S0 , S1 , S2 , . . . , Sn .
4. The combined discrete summation S = { S0 + S1 + S2 + . . . + Sn } must
discrete
also be something finite.
We may relax the first condition be letting f(x) be discontinuous at COUNTABLY many
points. But at each such point of discontinuity f(x) must have 2 definite values.
discontinuity
With this in mind let us review the example of the bouncing ball .
bouncing
ver
y(t) = v er tical position n
t0 t1 t2 ver
y ’(t) = ver tical speed
171 t3 t At each of the instants t0, t1, t2, . . . , y’(ti ) is discontinuous and has 2 definite
discontinuous
+
−
values y’(ti ) and y’(t i ). The ball bounces indefinitely. So there are infinitely many
such instants of discontinuity. However, these instants are COUNTABLE. So we may
integrate y’(t) over each of the subintervals (t0, t1), (t1, t2), (t2, t3), ... to get something
finite. (In this example the integration of y’(t)over each subintervals is zero, because
as we can see the CHANGE in height y(t) over each subintervals is zero. But this
may not generally be the case). The continuous summation will CONVERGE.
continuous
This means that y(t) will be something definite. From a practical or Physics p oint
P hysics
ertical
of view we know the ver tical position o f the ball at any instant.
THEORETICAL CONVERGENCE from the Physics point of view means that the ball will
come to rest. And at each bounce the ball loses a little energy . There are two
e nergy
ways in which the ball may come to rest.
1. The ball may bounce FINITELY many times and then come to rest in a FINITE
time interval [0, T]. Correspondingly, y ’ (t) will have only FINITELY many points
of discontinuity.
2. The ball may bounce indefinitely and “eventually or “in the LIMIT” the ball
eventually”
in
LIMIT
eventually
will come to rest. Here too there are two possibilities.
(i) The ball may bounce ind...
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 Fall '09
 TAMERDOğAN
 Limit, Δx

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