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Understanding Continuity in Functions: Concepts and Examples

Learn about continuous and discontinuous graphs, jump and removable discontinuities, infinite asymptotes, and how to determine points of discontinuity. Practice identifying discontinuity types and ensuring functions are continuous. Explore properties of continuity and apply them in examples.

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Understanding Continuity in Functions: Concepts and Examples

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  1. 14.4 - Continuity

  2. 14.4 - Continuity

  3. 14.4 - Continuity

  4. CONTINUITY SECTION 14.4 14.4 - Continuity

  5. Key Terms A. Continuous graph is where f(x) is at x = c if and only if there are no holes, jumps, skips or gaps in the graph B. To define a Continuous graph: • f(c) is defined • exists 14.4 - Continuity

  6. Continuous Functions 14.4 - Continuity

  7. Discontinuity Functions Jump Discontinuity Removable Discontinuity Infinite Asymptote • Graphs are separated under two categories: • Removable Discontinuityis where the graph is removable if the function can be made continuous by appropriately defining f(c) such as a hole • Non-Removable Discontinuity is where • Jump Discontinuity is where the graphs are separated from a graph • Infinite Asymptote is where the function approaches towards infinity 14.4 - Continuity

  8. Example 1 Determine the points at which the function discontinuous and identify the type of discontinuity Jump Discontinuity at x = 1 Removable Discontinuity at x = 2 14.4 - Continuity

  9. Example 2 Determine the points at which the function discontinuous and identify the type of discontinuity 14.4 - Continuity

  10. Your Turn Determine the points at which the function discontinuous and identify the type of discontinuity 14.4 - Continuity

  11. Example 3 Determine the points at which the function discontinuous and identify the type of discontinuity for 14.4 - Continuity

  12. Your Turn Determine the points at which the function discontinuous and identify the type of discontinuity for 14.4 - Continuity

  13. Example 4 Determine the value of c such that the function is continuous on the entire line for 14.4 - Continuity

  14. Example 5 Determine the value of c such that the function is continuous on the entire line for 14.4 - Continuity

  15. Your Turn Determine the value of c such that the function is continuous on the entire line for 14.4 - Continuity

  16. Continuity • Since continuity is considered using limits, the properties of limits carry into continuity. • Properties of Continuity (f and g are continuous at x = c) • Sum and Difference: f + g • Product: f(g) • Quotient: f/g ; g(c) ≠0 14.4 - Continuity

  17. Example 6 Use the definition of continuity and the properties of limits to show that f(x) = sin x + x3 + 5x – 2 when x = 0. 14.4 - Continuity

  18. Example 7 Use the definition of continuity and the properties of limits to show that when x = –2. 14.4 - Continuity

  19. Your Turn Use the definition of continuity and the properties of limits to show that when x = –2. 14.4 - Continuity

  20. Assignment Worksheet 14.4 - Continuity

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