What must be true for a function to be continuous on an interval of the form (a, b]?
Answer
Answer. We tweak the definition of continuous to involve the left-sided limit at x = b. f must be continuous on (a, b) and
f(b) must exist.
must exist.
f(b) must equal .
Example 2
What must be true for a function to be continuous on an interval of the form [a, b]?
Answer
f must be continuous on both [a, b) and on (a, b]. Equivalently, f must be continuous on (a, b) and
f(a) and f(b) must exist
and must exist
f(a) must equal and f(b) must equal
To think of this with more words and fewer expressions, the value of f at each endpoint must be what we'd expect if we let x approach the endpoint from within the interval.
Example 3
Determine if f is continuous on each of the following intervals.
[a, b]
[a, b)
(a, b]
(a, b)
Answer
[a, b]: No, because f(a) and f(b) are undefined.
[a, b): No, because f(a) is undefined.
(a, b]: No, because f(b) is undefined.
(a, b) Yes, because we don't need the consider the endpoints.
Example 4
Determine if f is continuous on each of the following intervals.
[a, b]
[a, b)
(a, b]
(a, b)
Answer
[a, b]: No, because f(b) is undefined.
[a, b): Yes, because and f is continuous on (a,b).
(a, b]: No, because f(b) is undefined.
(a, b): Yes, because we have a nice, smooth curve between these points.
Example 5
Determine if f is continuous on each of the following intervals.
[a, b]
[a, b)
(a, b]
(a, b)
Answer
Nope on all accounts, because f is not continuous on the open interval (a, b) to begin with.
Example 6
Determine if the function
is continuous on each of the following intervals.
[-1, 0]
[-1, 0)
(-1, 0]
(-1, 0)
[0, 1]
[0, 1)
(0, 1]
(0, 1)
Answer
If we draw the function, we get something like this:
This makes it easier to see what's going on. Since f(0) disagrees with both and , f will be discontinuous on any interval containing 0, and continuous on any other interval. Here's the breakdown:
[-1, 0] - No
[-1, 0) - Yes
(-1, 0] - No
(-1, 0) - Yes
[0, 1] - No
[0, 1) - No
(0, 1] - Yes
(0, 1) - Yes
Example 7
Determine if the function
is continuous on each of the following intervals.
[0, 1]
[0, 1)
(0, 1]
(0, 1)
[1, 5]
[1, 5)
(1, 5]
(1, 5)
Answer
Answer. If we graph the function, we see this:
Although f is discontinuous at 1 when we look at the whole graph, f(1) agrees with its right-sided limit (that is, ). This means if x = 1 is the left endpoint of an interval, f can be continuous on that interval.
[0, 1] - No, because .
[0, 1) - Yes, because x = 1 is not included in this interval.
(0, 1] - No, because .
(0, 1) - Yes, because x = 1 is not included in this interval.
[1, 5] - Yes, because f is continuous on (1, 5] and
[1, 5) - Yes, because f is continuous on (1, 5) and
(1, 5] - Yes, because f is continuous at all points greater than 1.
(1, 5) - Yes, because f is continuous at all points greater than 1.
must exist.
.
and 
must exist
and f(b) must equal 
and f is continuous on (a,b).
and 
, f will be discontinuous on any interval containing 0, and continuous on any other interval. Here's the breakdown:
). This means if x = 1 is the left endpoint of an interval, f can be continuous on that interval.
.
.
