There are no points in (2,5) where h is undefined, so we don't need to worry about that. The only point in (2,5) where the limit might be disagreeable or not exist is at x = 4. The left-hand limit is
The right-hand limit is
Since the one-sided limits disagree, doesn't exist, therefore h is discontinuous at x = 4. Therefore h is not continuous on the interval (2,5).
Determine whether f is continuous on each given interval.
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.