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Continuity

Continuity. (Section 2.6). Continuity. limit matches function value. Continuity checklist. Is the function value f(a) defined? Does the limit as xa exist? Does the limit match the function value?. A continuous function :

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Continuity

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  1. Continuity (Section 2.6)

  2. Continuity limit matches function value Continuity checklist Is the function value f(a) defined? Does the limit as xa exist? Does the limit match the function value? A continuous function: Graph has no holes, breaks, or jumps. You could draw it without lifting your pencil. When the x-values are close enough to each other, so are the corresponding function values. When all of the answers are YES, i.e. , we say f is continuous at a.

  3. Continuity limit matches function value Limit as x1 exists and matches function value, so function is continuous at x=1.

  4. Discontinuity limit does not match function value Does Not Exist Jump discontinuity (at x=2)

  5. Discontinuity limit does not match function value undefined Infinite discontinuity (at x=3)

  6. Discontinuity limit does not match function value undefined Removable discontinuity (at x=5) The discontinuity can be removed by defining f(5) to be 3.

  7. Discontinuity limit does not match function value undefined Removable discontinuity (at x=0) The discontinuity can be removed by defining f(0) to be 1.

  8. Discontinuity limit does not match function value Limit does not match function value The discontinuity can be removed by redefining f(1) to be 3. Removable discontinuity (at x=1)

  9. Discontinuity limit does not match function value Does Not Exist Does Not Exist Does Not Exist undefined There is no way to “repair” the discontinuity at x=0. Oscillating discontinuity (at x=0)

  10. Continuity limit matches function value Continuity checklist Is the function value f(a) defined? Does the limit as xa exist? Does the limit match the function value? A continuous function: Graph has no holes, breaks, or jumps. You could draw it without lifting your pencil. When the x-values are close enough to each other, so are the corresponding function values. If all of the answers are YES, i.e. , then f is continuous at a.

  11. Left and Right Continuity left-hand or right-hand limit matches function value Review: Circles Circle with radius 3, centered at origin: continuous at x = 1 Solve for y: left-continuous at x = 3 right-continuous at x = -3 Top half of circle: Bottom half of circle:

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