When looking at continuity on an open interval, we only care about the function values within that interval.
If we're looking at the continuity of a function on the open interval (a,b), we don't include a at b, they aren't invited. No value of x less than a or greater than b is invited, either. This is an exclusive club.
In a closed interval, denoted [a,b], we also must also invite our friends a and b to the pool party. Half-closed intervals either invite a, [a,b), or b, (a,b]. To talk about continuity on closed or half-closed intervals, we will see what this means from a continuity perspective. Start with a half-closed interval of the form [a,b). What does it mean for a function to be continuous on this interval?
Since we can only approach a from the right, we use the continuity definition for a right-sided limit instead of a two sided limit. We say f is continuous on [a,b) if f is continuous on (a,b) and
- f(a) exists
- exists
- f(a) and agree
Sample Problem
This function is continuous on the interval [a,b):
This function is continuous on (a,b). f(a) is defined, is defined, and the function value at a agrees with the right-sided limit at a.
Sample Problem
The following functions are not continuous on the interval [a,b).
This function is not continuous on [a,b) because f(a) is undefined. We could also say this function is not continuous on [a,b) because does not exist.
This function is not continuous on the interval [a,b) because .
This function is not continuous on the interval [a,b) because it is not continuous on the open interval (a,b).
Practice:
Let Determine whether f is continuous on each of the following intervals: | |
First, we will draw the function: Now it's easier to see what's going on. - [10,12] - f is not continuous on this interval because f(12) is undefined.
- [10,12) - f is continuous on this interval. For starters, f is continuous on (10, 12).
While it's true that if we look at the whole graph f is discontinuous at x = 10, for the sake of continuity on [10,12) we only care about the right-sided limit. Since , f is continuous on [10,12).
- (10,12] - f is not continuous on this interval because f(12) is undefined.
- (10,12) - f is continuous on this interval.
| |
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
- must exist.
- f(b) must equal .
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
- 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.
Determine if f is continuous on each of the following intervals.
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)
Determine if f is continuous on each of the following intervals.
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)
Determine if f is continuous on each of the following intervals.
Answer
No to all, because f is not continuous on the open interval (a,b).
Determine if the function
is continuous on each of the following intervals.
Answer
If we draw the function, we see 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.
- [-1,0] - No
- [-1,0) - Yes
- (-1,0] - No
- (-1,0) - Yes
- [0,1] - No
- [0,1) - No
- (0,1] - Yes
- (0,1) - Yes
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
- (1,5) - Yes