What does continuity at one point require?
Quick answer
A function is continuous at a point a when three things are true: f(a) is defined, the limit of f(x) as x approaches a exists, and that limit equals f(a). Check them in that order. For f(x)=x² at a=2, the function value is 4 and the limit is 4, so the function is continuous there. [S1]
The question, the details, the next step.
What exactly am I checking when a question asks whether a function is continuous at one point?
Check three conditions at the point, in order:
- The function has a value there: f(a) is defined.
- The limit as x approaches a exists.
- That limit equals the function value: lim(x→a) f(x) = f(a).
All three must hold for continuity at a. If the first check fails, stop: there is no defined function value to compare. If the limit does not exist, stop there. Otherwise, compare the limit and the function value. [S1]
The order is practical: first find what the function actually assigns at the point, then determine what values near the point approach, and finally compare.
Can you apply the checks to a simple function so I can see the steps?
Use f(x)=x² and test continuity at a=2.
- Is f(2) defined? Yes. Substitute 2: f(2)=2²=4.
- Does the limit exist? Yes. As x approaches 2, x² approaches 4, so lim(x→2) x²=4.
- Do they agree? Yes. The limit is 4 and f(2)=4.
Because all three checks pass, f(x)=x² is continuous at 2. [S1]
A compact written solution is: “f(2)=4; lim(x→2) x²=4; therefore lim(x→2) f(x)=f(2), so f is continuous at 2.” This makes the evidence for each condition visible rather than jumping straight to the conclusion.
How do I decide whether the limit exists, rather than just calculate the function's value?
Keep the two questions separate. f(a) asks for the function's assigned output at exactly a. The limit asks what f(x) approaches as x gets close to a. Finding the output at a does not, by itself, settle whether that nearby-value limit exists or whether the two agree.
For a basic example like x² at 2, evaluating nearby inputs leads to values approaching 4; the limit is 4, matching the value at 2. If a problem calls for checking behavior from each side, compare the left-hand and right-hand limits: one-sided limits can be needed. [S1] If you cannot establish that the full limit exists, you cannot complete the equality check.
What should I write for a different function, and how can I avoid skipping a condition?
Use this fill-in template for the point a:
- Value: f(a) = [value]. It is [defined / undefined].
- Limit: lim(x→a) f(x) = [value / does not exist].
- Comparison: [limit] [equals / does not equal] f(a).
- Conclusion: f is [continuous / not continuous] at a because [state the first failed condition, or say all three hold].
For the worked example, the filled version is: “f(2)=4, so it is defined. lim(x→2) x²=4. The limit equals f(2), so f is continuous at 2.” [S1]
If a condition fails, name it explicitly. For instance, do not claim continuity merely because f(a) exists; the limit must also exist and agree with that value. This template helps you show the reasoning, not just the final yes or no.
A practical checklist
Write down the point a you are testing.
Evaluate f(a) and record whether it is defined.
Determine whether lim(x→a) f(x) exists; check one-sided limits if needed.
Compare the limit with f(a).
State the conclusion and identify any failed condition.
A few more useful answers
If f(a) is defined, does that guarantee continuity at a?
No. The limit as x approaches a must also exist and equal f(a). A defined function value alone is only the first check. [S1]
What if the limit exists but is different from f(a)?
The function is not continuous at that point, because the limit and the function value do not agree. [S1]
Do I always need to calculate left- and right-hand limits separately?
Not necessarily. Check them when the problem or the function's behavior makes them relevant; one-sided limits can be used when needed. [S1]
Sources
References used for this guide.
- [S1] OpenStax Calculus Volume 1, Section 2.4: Continuity openstax.org · Checked