Chapter 2 | Limits
179
2.4 | Continuity
Learning Objectives
2.4.1 Explain the three conditions for continuity at a point. 2.4.2 Describe three kinds of discontinuities. 2.4.3 Define continuity on an interval. 2.4.4 State the theorem for limits of composite functions. 2.4.5 Provide an example of the intermediate value theorem.
Many functions have the property that their graphs can be traced with a pencil without lifting the pencil from the page. Such functions are called continuous . Other functions have points at which a break in the graph occurs, but satisfy this property over intervals contained in their domains. They are continuous on these intervals and are said to have a discontinuity at a point where a break occurs. We begin our investigation of continuity by exploring what it means for a function to have continuity at a point . Intuitively, a function is continuous at a particular point if there is no break in its graph at that point. Continuity at a Point Before we look at a formal definition of what it means for a function to be continuous at a point, let’s consider various functions that fail to meet our intuitive notion of what it means to be continuous at a point. We then create a list of conditions that prevent such failures. Our first function of interest is shown in Figure 2.32 . We see that the graph of f ( x ) has a hole at a . In fact, f ( a ) is undefined. At the very least, for f ( x ) to be continuous at a , we need the following condition: i. f ( a )is defined.
Figure 2.32 The function f ( x ) is not continuous at a because f ( a ) is undefined.
However, as we see in Figure 2.33 , this condition alone is insufficient to guarantee continuity at the point a . Although f ( a ) is defined, the function has a gap at a . In this example, the gap exists because lim x → a f ( x ) does not exist. We must add another condition for continuity at a —namely, ii. lim x → a f ( x ) exists.
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