Determine whether the function is continuous on the given interval.
on (-3,-2)
Answer
Yes. f is only discontinuous when x = -1, which is not in the interval (-3,-2), therefore f is continuous on (-3,-2).
Example 2
Determine whether the function is continuous on the given interval.
on (0,3)
Answer
Yes, g is continuous on this interval. g is discontinuous when x = 3, but this value is not in the interval (0,3).
Example 3
Determine whether the function is continuous on the given interval.
on (3,5)
Answer
Yes, h is continuous on this interval. The only value we need to worry about is x = 4, but h is continuous there.
Example 4
Determine whether the function is continuous on the given interval.
on (2,4).
Answer
No. m is discontinuous at 3 because m(3) = 6 but .
Example 5
Determine whether the function is continuous on the given interval.
on (1, 3)
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).
Example 6
Let
Determine whether the function f is continuous on each interval.
(-10,0)
(0,1)
(0,2)
(1,10)
(1,2)
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.
on (0,3)
.
on (1, 3)
does not exist, and f is discontinuous at x = 0.