Continuity (Section 2.6). When all of the answers are YES, i.e., we say f is continuous at a. Continuity limit matches function value 1.Is the function.

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Presentation transcript:

Continuity (Section 2.6)

When all of the answers are YES, i.e., we say f is continuous at a. Continuity limit matches function value 1.Is the function value f(a) defined? 2.Does the limit as x  a exist? 3.Does the limit match the function value? Continuity checklist 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.

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

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

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

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

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

Removable discontinuity (at x=1) 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.

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

Continuity limit matches function value 1.Is the function value f(a) defined? 2.Does the limit as x  a exist? 3.Does the limit match the function value? Continuity checklist If all of the answers are YES, i.e., then f is continuous at a. 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.

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