Why the Type of Interval Matters
Continuity is tested point by point, but the kind of point matters. At an interior point of an interval, values of the function occur on both sides, so continuity is tested with a two-sided limit. At an included endpoint, the interval approaches from only one side, so the appropriate test is one-sided.
This distinction is essential when a problem asks whether a function is continuous on an interval. The question is not whether the function has a two-sided limit at every endpoint one could draw around the interval. It is whether the function is continuous at every point belonging to that interval, using the limits available from within it.
Continuity at a Point Within an Interval
A function \(f\) is continuous at a point \(c\) in its domain if the limit of \(f(x)\) as \(x\) approaches \(c\) exists and equals the function value:
For an interior point of an interval, the domain contains points of the interval both less than and greater than \(c\), at least sufficiently close to \(c\). The limit in this definition is therefore two-sided. In terms of one-sided limits, the two-sided limit exists and equals \(f(c)\) precisely when both one-sided limits exist and have that same value:
At an endpoint, however, one side of the interval is absent. Requiring a two-sided limit there would ask about function values from a side that the interval does not contain. Continuity on an interval therefore uses a one-sided limit at each included endpoint.
Tests for Open, Closed, and Half-Open Intervals
Let \(a<b\), and suppose \(f\) is defined at every point in the interval being considered. The tests are:
| Interval | Points to test | Endpoint conditions |
|---|---|---|
| \((a,b)\) | Every \(c\) with \(a<c<b\) | No endpoint tests; \(a\) and \(b\) are excluded. |
| \([a,b]\) | Every interior point and both endpoints | \(\lim_{x\to a^+}f(x)=f(a)\) and \(\lim_{x\to b^-}f(x)=f(b)\). |
| \([a,b)\) | Every interior point and \(a\) | \(\lim_{x\to a^+}f(x)=f(a)\); \(b\) is excluded. |
| \((a,b]\) | Every interior point and \(b\) | \(\lim_{x\to b^-}f(x)=f(b)\); \(a\) is excluded. |
On an open interval, continuity means continuity at every point inside it. On a closed interval, continuity means continuity at every interior point and the appropriate one-sided continuity at both endpoints. A half-open interval combines these requirements: test its included endpoint, but not its excluded one.
These definitions concern continuity on the interval. If a function is also defined outside the interval, its behavior outside does not change whether its restriction to the interval is continuous there.
Why the One-Sided Endpoint Test Works
The one-sided test is not a weaker substitute chosen for convenience. It expresses continuity relative to the points actually in the interval. Consider the left endpoint \(a\) of \([a,b]\). A continuity test at \(a\) compares \(f(x)\) with \(f(a)\) for \(x\) in the interval and close to \(a\). Such \(x\) satisfy \(x\ge a\), so they approach \(a\) from the right.
Near \(a\), the points of \([a,b]\) have \(x\ge a\), so only approach from the right is possible within the interval.
The values \(f(x)\) must approach the value \(f(a)\) as those interval points approach \(a\).
That condition is exactly \(\lim_{x\to a^+}f(x)=f(a)\). At the right endpoint \(b\), the interval points instead approach from the left, giving \(\lim_{x\to b^-}f(x)=f(b)\).
At an interior point \(c\), points of the interval approach from both sides. Requiring the corresponding function values to approach \(f(c)\) from both sides is exactly the two-sided continuity condition. Thus, the interval tests follow directly from asking whether the function values approach the function value using points that belong to the interval.
Worked Examples
Worked Example: Continuity on an Open Interval
Determine whether \(f(x)=x^2\) is continuous on \((-1,2)\).
Every point \(c\) in this open interval satisfies \(-1<c<2\), so each is an interior point. We must check two-sided continuity at an arbitrary such \(c\). By the limit laws for polynomials,
The function value is \(f(c)=c^2\), so \(\lim_{x\to c}f(x)=f(c)\). This holds for every \(c\in(-1,2)\), hence \(f\) is continuous on \((-1,2)\). The numbers \(-1\) and \(2\) are not in the interval, so no endpoint continuity tests are required.
Check: At any interior \(c\), the limit and function value are both \(c^2\), as required.
Worked Example: Continuity on a Closed Interval
Determine whether \(g(x)=x^2-x\) is continuous on \([-2,1]\).
For every interior point \(c\), polynomial limit laws give
Now test the included left endpoint using a right-hand limit:
The right-hand limit equals the endpoint value. At the included right endpoint, use a left-hand limit:
This endpoint test also passes. The function is continuous at every interior point and satisfies both endpoint conditions; therefore \(g\) is continuous on \([-2,1]\).
Check: The endpoint limits are \(6\) and \(0\), matching \(g(-2)\) and \(g(1)\), respectively.
Worked Example: Continuity on a Half-Open Interval
Determine whether \(h(x)=\sqrt{x+1}\) is continuous on \([-1,3)\).
The included left endpoint is \(-1\). Since the interval extends to its right, test the right-hand limit:
For every interior point \(c\in(-1,3)\), the expression \(c+1\) is positive. The square-root function is continuous for positive inputs, so
Thus the function is continuous at every interior point and at the included endpoint \(-1\). The point \(3\) is excluded, so no continuity condition at \(3\) is needed. Therefore \(h\) is continuous on \([-1,3)\).
Check: The only endpoint included in the interval is \(-1\), and the required right-hand limit equals \(h(-1)\).
Worked Example: Passing Endpoint Tests Does Not Guarantee Continuity
Define \(p\) on \([0,2]\) by
At the included left endpoint, values just to the right of \(0\) equal \(0\), and \(p(0)=0\). Therefore
At the included right endpoint, values just to the left of \(2\) equal \(1\), and \(p(2)=1\). Therefore
But \(1\) is an interior point, so it requires a two-sided test. For \(x<1\), \(p(x)=0\), while for \(x>1\), \(p(x)=1\). Hence
These one-sided limits are unequal, so the two-sided limit at \(1\) does not exist. Although both endpoint tests pass, \(p\) is not continuous on \([0,2]\), because it is not continuous at the interior point \(1\).
Check: The endpoint conditions hold, but the interior test fails because \(0\ne 1\).
Common Pitfall: Testing the Wrong Side
A frequent error is to use a two-sided limit at an included endpoint, or to check only the side from outside the interval. For example, at the left endpoint of \([a,b]\), the relevant condition is \(\lim_{x\to a^+}f(x)=f(a)\), not \(\lim_{x\to a^-}f(x)=f(a)\). The points just left of \(a\) are not in this interval.
Here is a concrete case where that distinction matters. Define \(q\) on \([0,1]\) by \(q(x)=\sqrt{x}\). At \(0\), the correct test is
A mistaken two-sided test might claim that continuity fails because \(\sqrt{x}\) is not real-valued for \(x<0\). That does not refute continuity on \([0,1]\): those values of \(x\) lie outside the interval, and the required right-hand limit exists and equals the function value. Always identify the interval first, then select the limit direction from the included side.
A Reliable Checklist
When deciding whether a function is continuous on an interval, proceed in this order:
- Confirm that the function is defined at every point of the interval.
- Identify which endpoints are included and which are excluded.
- At each interior point, use the two-sided continuity condition.
- At an included left endpoint, compare the right-hand limit with the function value.
- At an included right endpoint, compare the left-hand limit with the function value.
- Do not impose a continuity test at an excluded endpoint, and do not let passing endpoint tests replace the interior-point checks.
The central distinction is between a point’s position in the interval and the direction from which interval points can approach it. Interior points require two-sided limits; included endpoints require one-sided limits from within the interval.
Check Your Understanding
For each question, identify the points that require testing and the appropriate direction of each limit.
- For continuity on \((2,5)\), which points require continuity tests, and do either endpoints require one-sided tests?
- State the endpoint conditions for a function to be continuous on \([a,b]\).
- For continuity on \((a,b]\), which endpoint requires a one-sided test, and from which side?
- A function passes both endpoint tests on \([0,4]\) but has unequal left- and right-hand limits at \(x=2\). Is it continuous on the interval? Explain.
- If a function is defined beyond \(b\), must continuity on \([a,b)\) include a continuity test at \(b\)? Why or why not?
