Test continuity with numeric limit checks. Review left, right, and function values. Detect gaps, jumps, and matching behavior fast.
| Case | Left Expression | Right Expression | Defined Value | Point | Expected Outcome |
|---|---|---|---|---|---|
| Matching Piecewise | x^2 - 1 | 3*x - 1 | 3 | 2 | Continuous |
| Jump Break | x + 1 | x + 3 | 2 | 1 | Discontinuous |
| Removable Hole | (x^2 - 1)/(x - 1) | (x^2 - 1)/(x - 1) | 0 | 1 | Discontinuous |
| Single Expression Domain Issue | sqrt(x-4) | 4 | Continuous from domain start | ||
The calculator applies the standard continuity rule at a point c. A function is continuous at c when the left limit exists, the right limit exists, both limits are equal, and the function value at c matches that common limit.
Core rule: lim x→c− f(x) = lim x→c+ f(x) = f(c)
The tool estimates one-sided limits numerically with nearby sample points. It uses c − h for the left side and c + h for the right side. Then it compares both estimates and the defined function value using the selected tolerance.
Comparison logic:
Left sample = f(c − h)
Right sample = f(c + h)
Continuous when |Left − Right| ≤ tolerance and |Limit − f(c)| ≤ tolerance.
Choose the analysis mode first. Use piecewise mode when the left side, right side, and defined point value are different. Use single expression mode when one formula describes the whole function.
Enter the continuity point c. Add a tolerance and small sample distances. Smaller distances improve local checking, while the tolerance controls how strictly results are compared.
Type expressions with x as the variable. You can use operators such as +, -, *, /, and ^. You can also use functions like sin(x), cos(x), sqrt(x), log(x), exp(x), and abs(x).
Press the button to calculate. The result appears above the form. You can then export the result as CSV or use the PDF button to save a printable report.
Continuity checks help developers validate formulas before deployment. A broken equation can crash analytics logic. It can also create unstable outputs in dashboards, simulations, and pricing tools. This calculator helps verify if a function behaves smoothly at a selected point.
The tool compares left behavior, right behavior, and the actual function value. These three checks reveal whether the target point is stable. It can identify a continuous point, a removable gap, or a jump. That makes debugging faster and clearer.
Software teams often encode piecewise formulas for billing, scoring, thresholds, and rule engines. These formulas may switch behavior near a cutoff. If both sides do not align, the output may surprise users. Continuity analysis exposes the problem before release.
This calculator uses nearby sample values around the target point. That method is practical for application testing. It avoids heavy symbolic processing. It still gives a strong estimate for continuity when formulas are well formed and the sample distance is chosen carefully.
The result section shows the left limit approximation, right limit approximation, and function value. It also labels the point with a status and classification. Developers can use this summary in QA reviews, bug reports, and validation notes.
Use continuity testing when you build calculators, scripts, APIs, and decision systems. Test every important threshold. Try multiple distances and tolerances. Review domain restrictions such as division by zero, invalid logarithms, and square root limits. Small checks prevent costly production errors.
It means the left limit, right limit, and function value agree at the selected point within the chosen tolerance.
Numeric checks rarely match perfectly. Tolerance allows tiny differences while still treating values as practically equal.
Use piecewise mode when the function changes definition across the target point or when the defined value is entered separately.
It is a hole. The left and right limits match, but the function value is missing or different.
Yes. Enter formulas like (x^2-1)/(x-1). Domain errors may appear near division by zero points.
The calculator supports sin, cos, tan, sqrt, log, exp, abs, floor, ceil, asin, acos, and atan.
That usually means the formula breaks at or near the point. Common causes are zero denominators, invalid logs, or invalid roots.
Yes. Use the CSV button for tabular output. Use the PDF button to save a printable report from the browser.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.