Calculator
Example Data Table
These examples show the type of output this calculator can help you inspect.
| Case | Mode | Input | Check point | Expected observation |
|---|---|---|---|---|
| Polynomial | Single | x^2+3*x-1 |
x = 2 | Continuous, because polynomials are continuous everywhere. |
| Hole | Single | ((x^2)-4)/(x-2) |
x = 2 | Removable discontinuity if no point value is supplied. |
| Piecewise jump | Piecewise | x+1, x-1, point 5 |
x = 1 | Jump behavior, because the one-sided limits do not match. |
Formula Used
For a function to be continuous at x = c, three conditions should hold.
- f(c) exists.
- lim x→c− f(x) and lim x→c+ f(x) both exist.
- The shared limit equals f(c).
This calculator estimates continuity numerically with shrinking step sizes.
L ≈ average of f(c - h) for the smallest valid left probes
R ≈ average of f(c + h) for the smallest valid right probes
Continuous if |L - R| ≤ tolerance and |f(c) - L| ≤ tolerance
How to Use This Calculator
- Select Single function for one expression, or Piecewise for different left and right rules.
- Enter the check point x = c.
- Provide a tolerance and base step size.
- Fill the graph range and sample count for visualization.
- Click Check Continuity to generate the report above the form.
- Review the verdict, graph, limit ladder, and nearby sample table.
- Use Download CSV for table data and Download PDF for a clean report.
FAQs
1) What does this calculator actually check?
It checks continuity at a selected point using numerical left-hand samples, right-hand samples, and the function value at that point. It also plots the function across a chosen interval.
2) Does it prove continuity exactly?
No. It gives a numerical continuity check, not a symbolic proof. Smaller steps and sensible tolerances improve confidence, but sampled results should still be interpreted carefully.
3) When should I use piecewise mode?
Use piecewise mode when the rule changes at the chosen point. You can enter one formula for values left of c, another for values right of c, and an optional point definition.
4) Why does the report say removable discontinuity suspected?
That label appears when the one-sided limits agree numerically, but the point value is undefined or missing. This often happens with holes such as simplified rational expressions.
5) What syntax should I use for multiplication and powers?
Use explicit multiplication like 3*x and powers like x^2. Implicit multiplication, such as writing 3x, is not supported in this file.
6) Which functions are supported?
Common functions include sin, cos, tan, asin, acos, atan, sqrt, abs, exp, ln, log, floor, ceil, round, plus constants pi and e.
7) What does the interval scan note mean?
It is a heuristic warning based on sampled graph values. Large jumps or undefined blocks may suggest discontinuities, asymptotes, or graph clipping inside the chosen interval.
8) How do the export buttons work?
The CSV button downloads summary tables from the latest calculation. The PDF button creates a client-side PDF report from the visible results section.