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The Journey of Life and Mathematics: Embracing Continuity

Life is a continuous journey, where the experience itself holds greater value than any destination. Osho reminds us that the essence of life lies in the pilgrimage, a joyous movement through existence without a defined goal. Similarly, in mathematics, the concept of continuity is fundamental; a continuous function can be traced smoothly without interruption. Understanding the nuances of continuity, limits, and discontinuities enriches our comprehension of both life and mathematics, revealing the beauty of remaining present in our journey.

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The Journey of Life and Mathematics: Embracing Continuity

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  1. 2.3 Continuity Life is a continuity always and always.There is no final destination it is going towards.Just the pilgrimage, just the journey in itself is life,not reaching to some point,no goal - just dancing and being in pilgrimage,moving joyously, without bothering about any destination.- Osho

  2. 2 1 1 2 3 4 Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil. A function is continuous at a point if the limit is the same as the value of the function. This function has discontinuities at x=1 and x=2. It is continuous at x=0 and x=4, because the one-sided limits match the value of the function

  3. Removable Discontinuities: (You can fill the hole.) Essential Discontinuities: oscillating infinite jump

  4. has a discontinuity at . Write an extended function that is continuous at . Note: There is another discontinuity at that can not be removed. Removing a discontinuity:

  5. Note: There is another discontinuity at that can not be removed. Removing a discontinuity:

  6. Also: Composites of continuous functions are continuous. Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous. examples:

  7. Because the function is continuous, it must take on every y value between and . Intermediate Value Theorem If a function is continuous between a and b, then it takes on every value between and .

  8. F2 Since f is a continuous function, by the intermediate value theorem it must take on every value between -1 and 5. Therefore there must be at least one solution between 1 and 2. Use your calculator to find an approximate solution. Is any real number exactly one less than its cube? Example 5: (Note that this doesn’t ask what the number is, only if it exists.) 1: solve

  9. Graph: Note resolution. Graphing calculators can sometimes make non-continuous functions appear continuous. CATALOG F floor( This example was graphed on the classic TI-89. You can not change the resolution on the Titanium Edition. The calculator “connects the dots” which covers up the discontinuities.

  10. Graph: GRAPH Graphing calculators can make non-continuous functions appear continuous. CATALOG F floor( If we change the plot style to “dot” and the resolution to 1, then we get a graph that is closer to the correct floor graph. The open and closed circles do not show, but we can see the discontinuities. p

  11. Yes! Its your Fun and Happy Joy Joy Pleasure Time #3-30 (multiples of 3), 36, 39, 42, 43, 48 Write #45 as a Journal.

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