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Understanding Continuity in Functions: Theory & Practice

Learn the essential concepts of continuity in functions, including criteria for continuity, types of discontinuity, and rules for determining continuity. Practice examples to deepen your understanding.

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Understanding Continuity in Functions: Theory & Practice

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  1. Continuity 2.3

  2. Continuity at a point • A function is continuous at any point given three things • 1. The function exists • 2. The limit as x approaches that point exists • 3. The function = the limit at that point

  3. Continuous from the right/left • All of the same rules apply, but we only check the one sided limit in question

  4. Example • Is the given function continuous at x=1? • X=4? • X=5? • Why not?

  5. Continuity on an interval • To be continuous on an interval means that the function is continuous at every point on that interval.

  6. Continuity on an interval • The following types of functions are continuous everywhere on their domain: • Polynomials, rational functions, root functions, trig functions, exponential functions, and logarithmic functions

  7. Continuity rules • If f and g are continuous functions • f + g is continuous • f - g is continuous • f x g is continuous • f / g is continuous where g ≠ 0 • f(g) is continuous

  8. Example • Where are each of the following functions discontinuous

  9. Example • Show that the following function is continuous on the interval[-1, 1]

  10. Types of discontinuity • A point of discontinuity is called REMOVABLE if the limit exists • A point of discontinuity is called NONREMOVABLE if the limit does not exist

  11. Example • Find any points of discontinuity and tell whether they are removable or nonremovable

  12. Homework • Pg 84 #1-14, 19-24

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