# Calc02_3 - 2.3 Continuity Grand Canyon Arizona Photo by...

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2.3 Continuity Grand Canyon, Arizona Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2002

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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. A function is continuous at a point if the limit is the same as the value of the function. This function has discontinuities at x=1 and x=2. It is continuous at x=0 and x=4, because the one-sided limits match the value of the function 1 2 3 4 1 2
jump infinite oscillating Essential Discontinuities: Removable Discontinuities: (You can fill the hole.)

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Removing a discontinuity: ( 29 3 2 1 1 x f x x - = - has a discontinuity at . 1 x = Write an extended function that is continuous at . 1 x = 3 2 1 1 lim 1 x x x - - ( 29 ( 29 ( 29 ( 29 2 1 1 1 lim 1 1 x x x x x x - + + = + - 1 1 1 2 + + = 3 2 = ( 29 3 2 1 , 1 1 3 , 1 2 x x x f x x - - = = Note: There is another discontinuity at that can not be removed.
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