Continuity Section 2.3a.

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Presentation transcript:

Continuity Section 2.3a

Find the points at which the function f is continuous, and the points at which f is discontinuous. In general, a continuous function is a function whose outputs vary continuously with the inputs and do not “jump” from one value to another without taking on values in between. Graphically, any function f(x) whose graph can be sketched in one continuous motion without lifting the pencil is an example of a continuous function…

Find the points at which the function f is continuous, and the points at which f is discontinuous. The function f is continuous at every point in its domain [0,4] except at x = 1 and x = 2. Note the relationship between the limit of f and the value of f at each point of the function’s domain. Points at which f is continuous: At x = 0, At x = 4, At 0 < c < 4,

Find the points at which the function f is continuous, and the points at which f is discontinuous. The function f is continuous at every point in its domain [0,4] except at x = 1 and x = 2. Note the relationship between the limit of f and the value of f at each point of the function’s domain. Points at which f is discontinuous: At x = 1, DNE At x = 2, but At c < 0, c > 4, these points are not in the domain of f

Definition: Continuity at a Point Interior Point: A function is continuous at an interior point c of its domain if Exterior: A function is continuous at a left endpoint a or is continuous at a right endpoint b of its domain if or , respectively.

Types of Discontinuity Graph Removable Discontinuity – The function can be redefined at a single point in order to “remove” the discontinuity (the graph has a “hole” in it). (0,1) Removable discontinuity at x = 0

Types of Discontinuity Graph Jump Discontinuity – The one-sided limits of the function at the given point exist, but have different values. (0,1) Jump discontinuity at x = 0

Types of Discontinuity Graph Infinite Discontinuity – The function approaches positive or negative infinity as x approaches the given point. Infinite discontinuity at x = 0

Types of Discontinuity Graph Oscillating Discontinuity – The function oscillates with increasing frequency as x approaches the given point Oscillating discontinuity at x = 0

Continuous Functions A function is continuous on an interval if and only if it is continuous at every point on the interval. A continuous function is one that is continuous at every point of its domain. Ex: Consider the reciprocal function Does this function have any points of discontinuity? Yes  The function has an infinite discontinuity at x = 0 Is this a continuous function? Yes  The only point of discontinuity (at x = 0) is not in the domain of the function, so the function is continuous on its domain!!!

Theorem: Properties of Continuous Functions If the functions f and g are continuous at x = c, then the following combinations are continuous at x = c. 1. Sums: 2. Differences: 3. Products: 4. Constant Multiples: for any number k 5. Quotients: provided Theorem: Composite of Continuous Functions If f is continuous at c and g is continuous at f (c), then the composite g f is continuous at c.

Guided Practice For the following function, (a) find each point of discontinuity, (b) Which of the discontinuities are removable? Not removable? Give reasons for your answers. (a) Point of discontinuity: (b) Removable  reassign: The graph?