We're asked about so many intervals here, that at first we're just going to ignore all of them. Instead, we will answer this question: "At what values of *x* is the function *f* discontinuous?" - Where is
*f* undefined? The piecewise definition says what to do for *x* < 3 and *x* > 3, but doesn't say what to do at *x* = 3, therefore *f*(3) is undefined. Since *f*(3) is undefined, *f* is discontinuous at *x* = 3.
*f* is also undefined at *x* = 5, since when we try to evaluate *f*(5) we run into problems trying to divide by zero. Therefore *f* is also discontinuous at *x* = 5.
- Where do limits fail to exist?
The limit doesn't exist, but we already know *f* is discontinuous at *x* = 5. The only other places we need to worry about are *x* = 2 and *x* = 3. Since we already know *f* is discontinuous at *x* = 3, the only place we need to worry about is *x* = 2. - The left-hand limit is
- The right-hand limit is
Since the one-sided limits agree,
We haven't found any new values of *x* where *f* is discontinuous. - Where do limits disagree with function values?
We already know *f* is discontinuous at *x* = 3 and *x* = 5, therefore we don't need to worry about those.
When *x* = 2, we have *f*(2) = 4. This agrees with the limit therefore *f* is continuous at *x* = 2. To summarize, the only values at which *f* is discontinuous are *x* = 3 and *x* = 5. Now that we've done all the hard work, we're ready to answer the real question. For each interval, we check to see if 3 or 5 is in the interval. If the answer is yes, then *f* is discontinuous on that interval. If neither 3 nor 5 is in the interval, then *f* is continuous on that interval. - (1,2) - Yes,
*f* is continuous on this interval because neither 3 nor 5 is in the interval (1,2). - (1,3) - Yes,
*f* is continuous on this interval because neither 3 nor 5 is in the interval (1,3) (3 is an endpoint, but is not in the interval). - (4,7) - No,
*f* is not continuous on this interval because 5 is in the interval (4,7). - (-100,100) - No,
*f* is not continuous on this interval. The interval (-100,100) contains both 3 and 5. - (3,5) - Yes,
*f* is continuous on this interval. 3 and 5 are endpoints, but neither is in the interval.
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