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Understanding Continuity in Mathematics

Learn about continuity in functions, including intuitive definitions, strange functions, and conditions for continuity. Explore continuity on closed intervals and discover where different types of functions are continuous or discontinuous. Complete exercises in Lesson 3.2 on Page 175.

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Understanding Continuity in Mathematics

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  1. Continuity Lesson 3.2

  2. Continuity in Context !!?? In this context, what do you think continuity means?

  3. Intuitive Look at Continuity • A function withoutbreaks orjumps • The graph can bedrawn without lifting the pencil 

  4. Strange Functions • A jump in the graph • But it is a function • The limit at 3 does not equal the value of the function at 3 • The function does not exist at x = 0

  5. Definition of Continuity at x = c • Three conditions required • f(c) is defined • exists • Which condition is violated by each of our strange functions?

  6. Continuity on Closed Interval • Given closed interval [a, b] … • f(x) continuous if • Continuous on the open interval (a, b) • Continuous from right at x = a • Continuous from left at x = b • Example

  7. Where Are They Continuous? • Polynomial Functions • Rational Functions • Root Functions • Exponential Functions • Logarithmic Functions

  8. Where Are They Continuous? • Try it out … where are these functions continuous? • Where are they discontinuous

  9. Assignment • Lesson 3.2 • Page 175 • Exercises 1 – 27 odd

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