Sec 2.5: Continuity Continuous Function

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Presentation transcript:

Sec 2.5: Continuity Continuous Function Intuitively, any function whose graph can be sketched over its domain in one continuous motion without lifting the pencil is an example of a continuous function.

Sec 2.5: Continuity Continuity at a Point (interior point) Continuity Test A function f(x) is continues at a point a if Example: study the continuity at x = -1

Sec 2.5: Continuity Continuity at a Point (interior point) Continuity Test A function f(x) is continues at a point a if Example: study the continuity at x = 4

Sec 2.5: Continuity Continuity at a Point (interior point) Continuity Test A function f(x) is continues at a point a if Example: study the continuity at x = 2

Sec 2.5: Continuity Continuity at a Point (interior point) Continuity Test A function f(x) is continues at a point a if Example: study the continuity at x = -2

Sec 2.5: Continuity Cont a Continuity at a Point (end point) Cont from A function f(x) is continues at an end point a if Cont from left at a Cont from right at a

Types of Discontinuities. Sec 2.5: Continuity Types of Discontinuities. removable discontinuity Which conditions infinite discontinuity Later: oscillating discontinuity: jump discontinuity

Sec 2.5: Continuity

Sec 2.5: Continuity Exam1-102

Sec 2.5: Continuity

Sec 2.5: Continuity Exam1-122

Sec 2.5: Continuity Continuous on [a, b]

Sec 2.5: Continuity Remark The inverse function of any continuous one-to-one function is also continuous.

Sec 2.5: Continuity Inverse Functions and Continuity The inverse function of any continuous one-to-one function is also continuous. This result is suggested from the observation that the graph of the inverse, being the reflection of the graph of ƒ across the line y = x

Sec 2.5: Continuity

Sec 2.5: Continuity continuous

Sec 2.5: Continuity Exam1-101

Sec 2.5: Continuity Geometrically, IVT says that any horizontal line between ƒ(a) and ƒ(b) will cross the curve at least once over the interval [a, b].

Sec 2.5: Continuity The Intermediate Value Theorem y0=ƒ(c) 1) ƒ(x) cont on [a,b] 2) y0 between ƒ(a) and ƒ(b) c in [a,b]

Sec 2.5: Continuity One use of the Intermediate Value Theorem is in locating roots of equations as in the following example.

Sec 2.5: Continuity E1 TERM-121

Sec 2.5: Continuity Exam1-101

Sec 2.5: Continuity