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Continuity - CONTINUITY We have seen that the limit of a...

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CONTINUITY We have seen that the limit of a function as x approaches a can sometimes be found by calculating the value of the function at x = a . Functions with this property are called continuous at a . Mathematical definition is as follows. Continuity A function f is continuous at a if lim x a f ( x ) = f ( a ) . In fact, this definition requires three conditions of continuity. Those are 1. Existence of f ( a ), 2. Existence of the limit lim x a f ( x ), and 3. lim x a f ( x ) = f ( a ) . If any one of the above is not fulfilled, we say that f is discontinuous at a . f is continuous on an interval if it is continuous at every point on the interval. We can also define continuity from the left(right) by using the left(right)-hand limit. Example 1. Find the points at which each of the following functions are dis- continuous. (a) f ( x ) = x 3 - 1 x - 1 (b) f ( x ) = H ( x ) (c) f ( x ) = ( x 2 + x + 1 if x 6 = 1 2 if x = 1 . Solution In (a), f (1) does not exist, thus it is discontinuous at 1.
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