1.5 Continuity. Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without.

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Presentation transcript:

1.5 Continuity

Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil. Definition: A function f is continuous at a number a if (the limit is the same as the value of the function). This function is discontinuous (has discontinuities) at x=1 and x=2. It is continuous everywhere else on the interval [0,4]

jump infinite oscillating Essential Discontinuities: Removable Discontinuities: (You can fill the hole.)

Definition: A function f is continuous from the right at a number a if and f is continuous from the left at a number a if Definition: A function f is continuous on an interval if it is continuous at every number in the interval. Examples on the board.

Theorem: If f and g are continuous at a and c is a constant, then the following functions are also continuous at a: Theorem: The following types of functions are continuous at every number in their domains: polynomials, rational functions, root functions, trigonometric functions. Example: The function is continuous on the intervals [0,2) and (2,  )

Intermediate Value Theorem If a function is continuous between a and b, then it takes on every value between and. Because the function is continuous, it must take on every y value between and. Examples on the board.