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Continuity We have a good general idea of what it means for a graph to be continuous . Our purpose today is to work with a more formal definition of continuity, and to see how it relates to our more natural, gut instinct, definition. The first potential issue we need to address is that a function is not said to be continuous or discontinuous as a whole. Rather, the function is described as continuous or discontinuous at certain x values. The function to the left, g ( x ), is discontinuous at x = 3. g ( x ) is said to be continuous on the intervals (-∞,3) and (3,∞). We are already familiar with various types of discontinuities. This one is called a removable discontinuity .
Are the following graphs continuous or discontinuous as x approaches c ? If they are discontinuous, describe the discontinuity.
If you recall the cartoon that we watched about limits, you might remember that the narrator referred to the left and right hand limits. These were symbolized like this: The left and right hand limits refer to the behavior of the function as x -> c from the negative and positive sides. English translation: a function only has a limit as x approaches c if it is approaching the same value from the left of c as it is from the right of c .
A Formal Definition of Continuity A function f is continuous at x = c if and only if:
<ul><li>For the function: </li></ul><ul><li>Sketch the graph of f ( x ) for k = 1. </li></ul><ul><li>What type of discontinuity does the graph show at x = 2? </li></ul><ul><li>Find the value of k that makes f continuous at x = 2. </li></ul>