Section 2.4
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This document discusses continuity and discontinuity in functions. It defines a function as continuous at a point c if the left and right hand limits are the same as x approaches c. A function is discontinuous if the left and right limits do not match up. Examples are given of different types of discontinuities like removable discontinuities.
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Section 2.4
- 1. 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 .
- 2. Are the following graphs continuous or discontinuous as x approaches c ? If they are discontinuous, describe the discontinuity.
- 3. 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 .
- 4. A Formal Definition of Continuity A function f is continuous at x = c if and only if:
- 5. For the function: Sketch the graph of f ( x ) for k = 1. What type of discontinuity does the graph show at x = 2? Find the value of k that makes f continuous at x = 2.