3/9/2023 0 Comments Piecewise function practiceĪ function is continuous over an open interval if it is continuous at every point in the interval. Ī function f ( x ) f ( x ) is said to be continuous from the left at a if lim x → a − f ( x ) = f ( a ). These examples illustrate situations in which each of the conditions for continuity in the definition succeed or fail.Ī function f ( x ) f ( x ) is said to be continuous from the right at a if lim x → a + f ( x ) = f ( a ). The next three examples demonstrate how to apply this definition to determine whether a function is continuous at a given point. 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.
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