Let us compare two examples of discrete summation and

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Unformatted text preview: An each such point of discontinuity f(x) has 2 definite values. discontinuity 3. Over each of the (n + 1) sub-intervals 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 sub-intervals (t0, t1), (t1, t2), (t2, t3), ... to get something finite. (In this example the integration of y’(t)over each sub-intervals is zero, because as we can see the CHANGE in height y(t) over each sub-intervals 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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