Calculator Inputs
Use cubic expressions on each side of the boundary point. This is useful for testing branch behavior, feature toggles, and numeric rule transitions.
Plotly Graph
The chart separates the left-hand rule, right-hand rule, limit markers, and the stored point value. It helps reveal holes and jumps instantly.
Example Data Table
| Case | Boundary c | Left expression | Right expression | f(c) | Expected status |
|---|---|---|---|---|---|
| Shared limit with correct value | 2 | x² - 4x + 5 | x² - 4x + 5 | 1 | Continuous |
| Shared limit with wrong value | 2 | x² - 4x + 5 | x² - 4x + 5 | 3 | Removable discontinuity |
| No stored point | 2 | x² - 4x + 5 | x² - 4x + 5 | Undefined | Removable discontinuity |
| Different one-sided limits | 2 | x² - 4x + 5 | x² - 3x + 5 | 1 | Jump discontinuity |
Formula Used
Left-hand limit: L = lim x→c⁻ f(x), computed by evaluating the left cubic at x = c.
Right-hand limit: R = lim x→c⁺ f(x), computed by evaluating the right cubic at x = c.
Two-sided limit exists when: |L - R| ≤ ε
Continuity at c requires: L = R and f(c) = L, within the chosen tolerance ε.
Polynomial model: f(x) = ax³ + bx² + cx + d
How to Use This Calculator
- Enter the boundary point where you want to test continuity.
- Provide cubic coefficients for the left-side rule and right-side rule.
- Set a tolerance that matches your precision needs.
- Choose whether the function is defined at the boundary point.
- Enter the stored boundary value when it exists.
- Set graph range and sample count for visualization.
- Press Calculate Continuity to show results above the form.
- Use the CSV and PDF buttons to export your analysis.
FAQs
1) What does this calculator test?
It checks whether a function is continuous at one boundary point. It compares left and right limits, evaluates the stored point value, and classifies the result as continuous, removable, or jump-based.
2) Why is this useful in software development?
Many systems switch rules at thresholds. This calculator helps validate boundary behavior in pricing engines, risk scoring, simulations, interpolators, and feature-flag logic where abrupt numeric changes can break expected behavior.
3) What is a removable discontinuity?
A removable discontinuity happens when both one-sided limits agree, but the point value is missing or incorrect. The graph usually shows a hole or a misplaced dot at the boundary.
4) What is a jump discontinuity?
A jump discontinuity occurs when the left-hand and right-hand limits are different. This often means two rules meet at the same threshold but produce conflicting outputs.
5) Why does tolerance matter?
Floating-point arithmetic can create tiny numerical differences. Tolerance lets you decide when two values are practically equal, which is important for automated tests and approximate models.
6) Can I use this for non-cubic formulas?
Yes. Set unused higher-order coefficients to zero. For example, a quadratic only needs x², x, and constant terms. A linear rule only needs x and constant values.
7) How should I interpret the graph?
The left trace shows values just before the boundary. The right trace shows values just after it. Separate markers show the left limit, right limit, and stored point value.
8) How do I make a discontinuous point continuous?
If both limits already match, set the boundary value equal to that shared limit. If the limits differ, adjust one or both formulas so they converge at the boundary.