![]() If lim x → a f ( x ) = f ( a ), lim x → a f ( x ) = f ( a ), then the function is continuous at a. ![]() If lim x → a f ( x ) ≠ f ( a ), lim x → a f ( x ) ≠ f ( a ), then the function is not continuous at a. Compare f ( a ) f ( a ) and lim x → a f ( x ).If lim x → a f ( x ) lim x → a f ( x ) exists, then continue to step 3. If lim x → a f ( x ) lim x → a f ( x ) does not exist (that is, it is not a real number), then the function is not continuous at a and the problem is solved. In some cases, we may need to do this by first computing lim x → a − f ( x ) lim x → a − f ( x ) and lim x → a + f ( x ). If f ( a ) f ( a ) is defined, continue to step 2. If f ( a ) f ( a ) is undefined, we need go no further. Check to see if f ( a ) f ( a ) is defined.Problem-Solving Strategy: Determining Continuity at a Point At the very least, for f ( x ) f ( x ) to be continuous at a, we need the following condition: We see that the graph of f ( x ) f ( x ) has a hole at a. Our first function of interest is shown in Figure 2.32. We then create a list of conditions that prevent such failures. Continuity at a Pointīefore 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. Intuitively, a function is continuous at a particular point if there is no break in its graph at that point. We begin our investigation of continuity by exploring what it means for a function to have continuity at a point. They are continuous on these intervals and are said to have a discontinuity at a point where a break occurs. Other functions have points at which a break in the graph occurs, but satisfy this property over intervals contained in their domains. Many functions have the property that their graphs can be traced with a pencil without lifting the pencil from the page. ![]()
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