Calculus I Chapter 2(6) Continuity

1. Calculus IChapter 2(6)Continuity

2. Limits: Piece Functions = -2 (top) = 2 (middle) = -4 (Middle) = 2 (bottom) = No Limit (not equal) = 2 (are equal)

3. Limits: Piece Functions = 3 = 0 = No Lim = 0 = No Limit = 0

4. Continuity at a point Three things can happen at a point on a curve: The curve is continuous Solid – no breaks The curve is missing a single point The curve has a “vertical break”

5. Continuity at a point Mathematically limits and function definitions are involved in each case. The curve is continuous The defined point Equals the limit

6. The curve is missing a single point The defined point Does not equal the limit The curve has a “vertical break” There is no limit

7. Requirements for Continuity The function must be defined at the point The limit must exist The limit = the definition

8. GraphingExample At x = 2 Definition: Limit: Equal: At x = 1 Definition: Limit Equal: At x = 4 Definition: Limit: Equal: 0 0 1 -1 None 0 No Not Continuous No Not Continuous 0 Continuous

9. Algebraic Functions and Continuity All polynomials are continuous Two Trig functions are continuous Sine Cosine Fractions are continuous where the denominator ≠ 0 Radicals are continuous where the inside >0 Piece functions must be checked where the “breaks” are.

10. Types of discontinuities Removable – only a single point is missing Non-Removable – there is no limit at the point (there is a jump between the two sides)

11. Give the points (if any) where the function is not continuous. This is not continuous at because of Tangent These are non-removable This is not continuous on 0 < x < 2 No Yes Yes This is non-removable 0 2

12. At x = 3 Definition: Limit: Equal: At x = 1 Definition: Limit: Equal: At x = -3 Definition: Limit Equal: Not 3 0 0 No Lim 0 No Not Continuous (removable by adding one point) Yes Continuous No Not Continuous (non-removable)

13. Find any discontinuities x = -3 is a non-removable discontinuity x = 2 is a removable discontinuity