Continuity on an Interval via Formulas Examples

Let 

Is h continuous on the interval (2,5)?

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).

Let 

Determine whether f is continuous on each given interval.

• (1,2)
• (1,3)
• (4,7)
• (-100,100)
• (3,5)

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.