Limits and Continuity Unit 1 Day 4.

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Presentation transcript:

Limits and Continuity Unit 1 Day 4

Continuity at a POINT A function is continuous at a if lim 𝑥→𝑎 𝑓 𝑥 =𝑓(𝑎) There are 3 important parts to this definition: f(a) exists The lim 𝑥→𝑎 𝑓 𝑥 exists (which means lim 𝑥→ 𝑎 − 𝑓 𝑥 = lim 𝑥→ 𝑎 + 𝑓 𝑥 ). The two values above are equal. A function is considered continuous if it is continuous at every point in its domain.

Continuity Example Let 𝑓 𝑥 = 4𝑥+1 . Show that f(x) is continuous at x = 2. f 2 = 4 2 +1 =3 lim x→2 4x+1 = 4 2 +1 =3 Since lim 𝑥→2 𝑓 𝑥 =3=𝑓(2) , f(x) is continuous at x = 2.

Continuity

Types of Discontinuities Non - Removable Discontinuities Removable Discontinuity Jump Discontinuity Infinite (Asymptotic) Discontinuity Oscillating Discontinuity

Types of Discontinuities Non - Removable Discontinuities Removable Discontinuity move the point and the function will be continuous Jump Discontinuity Infinite (Asymptotic) Discontinuity Oscillating Discontinuity

Types of Discontinuities Non - Removable Discontinuities Removable Discontinuity Now continuous Jump Discontinuity Infinite (Asymptotic) Discontinuity Add a point and the function will be continuous Oscillating Discontinuity

Types of Discontinuities Non - Removable Discontinuities Removable Discontinuity Now continuous Jump Discontinuity Infinite (Asymptotic) Discontinuity Now continuous Oscillating Discontinuity

Types of Discontinuities Non - Removable Discontinuities Removable Discontinuity Jump Discontinuity Infinite (Asymptotic) Discontinuity Oscillating Discontinuity

Continuity Examples Determine where the discontinuities are and classify them as removable, infinite, jump, or oscillating. Then state the interval on which the function is continuous. Discontinuities Continuous No discontinuities Continuous on (-∞, ∞ ) Jump discontinuity at x = 0 Continuous on (-∞, 0) U [0, ∞ ) Removable discontinuity at x = -5 Infinite discontinuity at x = -1 Continuous on (−∞, -5) u (-5, -1) U (-1, ∞) Infinite Discontinuity at x = -3 Continuous on (-∞, -3) U (-3, ∞ ) x ≠ 0 Removable discontinuity at x = 0 Continuous on (-∞, 0) U (0, ∞ )

Intermediate Value Theorem What was a speed you are 100 % sure you must have gone in the time in between? Why? What was a speed that you could have gone in between but you aren’t so sure? Why?

Intermediate Value Theorem What was a price you are 100 % sure the iphone must have been in between? Why? What was a price that the iphone might have had in between but you aren’t 100% sure? Why?

Intermediate Value Theorem The Intermediate Value Theorem states that if f(x) is continuous in the closed interval [a,b] and f(a) M f(b), then at least one c exists in the interval [a,b] such that: f(c) = M

IVT Example Let 𝑓 𝑥 = 𝑥 3 −𝑥−1. Show that f(x) has a zero. f 1 =−1 f(x) is continuous on [1,2] so by the IVT, there must be a value on [1,2] where f(x) = 0.