Chapter 1: Limits
Section 1.6: Continuity
Example 1.6.1
Show that the function fx={sinxxx≠01x=0 is continuous at x=0.
Solution
In Figure 1.6.1(a), the graph of sinx/x shows there is a "hole" at the point 0,1 because this expression is not defined at x=0. The function fx supplies the value 1 at x=0, so a graph of fx would not have a hole at this point.
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.
Figure 1.6.1(a) Graph of sinx/x
Expression palette: Limit template Context Panel: Evaluate and Display Inline
limx→0sinxx = 1
For the given function f, it is indeed true that limx→0fx=1=f0, so f is indeed continuous at x=1. This result is intuitively consistent with Figure 1.6.1(a) if the hole at 0,1 is filled in with the function value f0=1.
The function gx=sinx/x has a removable discontinuity at x=0 because it can be extended to a function f , whose domain includes x=0, in such a way as to make f continuous at x=0. If a function has a removable discontinuity, its graph can display the discontinuity via the Option "showremovable" in the plot command.
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