Continuity of Functions Exercises

Answer

Yes. f is only discontinuous when x = -1, which is not in the interval (-3,-2), therefore f is continuous on (-3,-2).

Answer

Yes, g is continuous on this interval. g is discontinuous when x = 3, but this value is not in the interval (0,3).

Answer

Yes, h is continuous on this interval. The only value we need to worry about is x = 4, but h is continuous there.

Answer

No. m is discontinuous at 3 because m(3) = 6 but .

Answer

Yes. The denominator factors as (x +2)(x + 1) so the function is undefined at x = -2 and x = -1. These are the only points of discontinuity, but neither is in the interval (1, 3).

Answer

Since we're asked about the same function on so many intervals, first we'll figure out all the values at which f is discontinuous.

• Where is f undefined?
• f is undefined at x = 1 because the function definition forgot to say what to do when x = 1. f is also undefined at 2, since  is undefined at 2.

• Where does the limit of f not exist?
• We already know f is undefined at x = 1 and x = 2, therefore we don't need to worry about those values. Since x = 0 is a spot where the definition changes, we'd better check out the limit there.
The left-hand limit is 

The right-hand limit is 

Since the one-sided limits disagree, does not exist, and f is discontinuous at x = 0.

• There are no spots where f(x) exists, the limit exists, and they disagree. We've already taken care of all the possible trouble spots.

To summarize, f is discontinuous at x = 0, x = 1, and x = 2.Now we can answer the real questions.

• (-10,0): Yes, f is continuous on this interval since none of 0, 1, and 2 are in the interval.
• (0,1): Yes, f is continuous on this interval since none of 0, 1, and 2 are in the interval.
• (0,2): No, f is not continuous on this interval since 1 is in the interval.
• (1,10): No, f is not continuous on this interval since 2 is in the interval.
• (1,2): Yes, f is continuous on this interval since none of 0, 1, and 2 are in the interval.