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In Figure 1.6.1(a), the graph of shows there is a "hole" at the point because this expression is not defined at . The function supplies the value 1 at , so a graph of would not have a hole at this point.
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The calculus student is rarely asked to apply Definition 1.6.1 to show that a function is continuous. Generally, any demonstration of continuity is confined to an application of the three items in Table 1.6.1.
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Figure 1.6.1(a) Graph of
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Expression palette: Limit template
Context Panel: Evaluate and Display Inline
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For the given function , it is indeed true that , so is indeed continuous at . This result is intuitively consistent with Figure 1.6.1(a) if the hole at is filled in with the function value
The function has a removable discontinuity at because it can be extended to a function , whose domain includes , in such a way as to make continuous at . If a function has a removable discontinuity, its graph can display the discontinuity via the Option "showremovable" in the plot command.
