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Presentation transcript:

5 4 2 1 5 7 -1

Definition: A function is continuous at a number a if: Continuity Definition: A function is continuous at a number a if:

Show that is continuous at a = 3

Continuity: 1. Function must exist 2. Limit must exist 3. f(a) = A function that is continuous has no holes, vertical asymptotes, or jumps. If they do the function is called discontinuous.

At what point(s) is f(x) discontinuous? Continuity At what point(s) is f(x) discontinuous? 5 4 2 1 5 7 -1

Types of discontinuous functions: 1. Removable (hole)

2. Infinite (asymptote)

3. Jump

4. Oscillating

Use the graph to answer the questions that follow (only #1, 2). -2 2

Continuous Functions A continuous function is one that is continuous at every point of its domain. Polynomials Rationals Absolute Value Exponential Logarithmic Trigonometric Radical *Algebraic combinations of continuous functions are continuous wherever they are defined* All composites of continuous functions are continuous. If f is continuous and g is continuous at f(c), then the composite g(f(x)) is continuous at c.

Intermediate Value Theorem A function is said to have the intermediate value property if it never takes on two values without taking on all the values in between.

Determine if this function has the intermediate value property.

Continuity Ex. 6: Sketch the graph of the function. Explain why the function is discontinuous at the given number a.

Sketch the graph of a function that has a jump discontinuity at x = 2 and a removable discontinuity at x = 4, but is continuous elsewhere.

Right / Left Hand Continuity Left Hand Continuity: Right Hand Continuity:

Determine if the graph is left/right continuous or neither at the following values: x = -2 ; x = 0 ; x = 2 ; x = 4 ; x = 6 -2 2

Sketch a possible graph for a function f that has the stated properties. f(4) exists, exists, but f is not continuous at x = 4

Find a value for a so that the function is continuous.

Find a value for a so that the function is continuous.

Using the graph of f(x), answer the questions that follow