Continuity TS: Making decisions after reflection and review.

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Presentation transcript:

Continuity TS: Making decisions after reflection and review

Objectives To find the intervals on which a function is continuous. To find the intervals on which a function is continuous. To find any discontinuities of a function. To find any discontinuities of a function. To determine whether discontinuities are removable or non-removable. To determine whether discontinuities are removable or non-removable.

Video Clip from Calculus-Help.com Calculus-Help.com Continuity

What makes a function continuous? Continuous functions are predictable… Continuous functions are predictable… 1) No breaks in the graph A limit must exist at every x -value or the graph will break. 2) No holes or jumps The function cannot have undefined points or vertical asymptotes.

Continuity Key Point: Key Point: Continuous functions can be drawn with a single, unbroken pencil stroke.

Continuity Mathematically speaking… Mathematically speaking… If f (x) is continuous, then for every x = c in the function, In other words, if you can evaluate any limit on the function using only the substitution method, then the function is continuous. In other words, if you can evaluate any limit on the function using only the substitution method, then the function is continuous.

Continuity of Polynomial and Rational Functions A polynomial function is continuous at every real number. A polynomial function is continuous at every real number. A rational function is continuous at every real number in its domain. A rational function is continuous at every real number in its domain.

Polynomial Functions Both functions are continuous on. Both functions are continuous on.

Rational Functions continuous on: continuous on: continuous on: continuous on:

Rational Functions continuous on: continuous on: continuous on: continuous on:

Piecewise Functions continuous on continuous on

Discontinuity Discontinuity: a point at which a function is not continuous Discontinuity: a point at which a function is not continuous

Discontinuity Two Types of Discontinuities Two Types of Discontinuities 1) Removable (hole in the graph) 2) Non-removable (break or vertical asymptote) A discontinuity is called removable if a function can be made continuous by defining (or redefining) a point. A discontinuity is called removable if a function can be made continuous by defining (or redefining) a point.

Two Types of Discontinuities

Discontinuity Find the intervals on which these function are continuous. Find the intervals on which these function are continuous. Point of discontinuity: Vertical Asymptote: Removable discontinuity Non-removable discontinuity

Discontinuity Continuous on:

Discontinuity

Discontinuity Determine the value(s) of x at which the function is discontinuous. Describe the discontinuity as removable or non-removable. Determine the value(s) of x at which the function is discontinuous. Describe the discontinuity as removable or non-removable. (A)(B) (C)(D)

Discontinuity (A) Removable discontinuity Non-removable discontinuity

Discontinuity (B) Removable discontinuity Non-removable discontinuity

Discontinuity (C) Removable discontinuity Non-removable discontinuity

Discontinuity (D) Removable discontinuity Non-removable discontinuity

Conclusion Continuous functions have no breaks, no holes, and no jumps. Continuous functions have no breaks, no holes, and no jumps. If you can evaluate any limit on the function using only the substitution method, then the function is continuous. If you can evaluate any limit on the function using only the substitution method, then the function is continuous.

Conclusion A discontinuity is a point at which a function is not continuous. A discontinuity is a point at which a function is not continuous. Two types of discontinuities Two types of discontinuities Removable (hole in the graph) Removable (hole in the graph) Non-removable (break or vertical asymptote) Non-removable (break or vertical asymptote)