Calculator Inputs
Example Data Table
| Function | Interval | ε | Tested δ | Sampled ω(δ) | Verdict |
|---|---|---|---|---|---|
| x^2 | [0, 1] | 0.2000 | 0.1000 | 0.1900 | Pass |
| sin(x) | [0, 6.283185] | 0.1500 | 0.1500 | 0.1494 | Pass |
| sqrt(x) | [0, 1] | 0.1000 | 0.0100 | 0.0999 | Pass |
Formula Used
A function f is uniformly continuous on an interval I if, for every ε > 0, there exists a δ > 0 such that whenever |x - y| < δ, we also have |f(x) - f(y)| < ε for all x, y ∈ I.
This calculator estimates the sampled modulus of continuity:
ω(δ) = max{|f(x) - f(y)| : |x - y| ≤ δ, x, y in the sampled interval}
For a chosen epsilon, a sampled delta is accepted when ω(δ) < ε.
It also computes a finite-difference Lipschitz estimate:
L ≈ max |f(xᵢ₊₁) - f(xᵢ)| / |xᵢ₊₁ - xᵢ|
If the function behaves like a Lipschitz function on the interval, then |f(x)-f(y)| ≤ L|x-y|, so a sufficient choice is:
δ ≤ ε / L
How to Use This Calculator
- Enter the function in terms of x, such as sin(x), x^2, or sqrt(x+1).
- Set the interval start and end values. This calculator is intended for bounded intervals.
- Enter the epsilon value you want to test.
- Enter a candidate delta to see whether the sampled ε-δ condition passes.
- Choose a sample count. Higher values improve resolution but increase computation.
- Choose delta grid steps to estimate how the sampled modulus changes across the interval.
- Click Analyze Uniform Continuity to generate results, tables, and the graph.
- Use the CSV and PDF buttons to export the report after calculation.
Frequently Asked Questions
1) What does uniform continuity mean?
Uniform continuity means one delta works across the whole interval for a chosen epsilon. The rule does not change from point to point.
2) How is uniform continuity different from ordinary continuity?
Ordinary continuity may use a different delta near each point. Uniform continuity requires a single delta that works everywhere on the interval.
3) Why are closed bounded intervals important?
A continuous function on a closed bounded interval is uniformly continuous by the Heine-Cantor theorem. That is why this calculator is most reliable on finite intervals.
4) Can this calculator prove uniform continuity?
No. It provides strong numerical evidence using sampling. A full proof still depends on the function’s analytic properties and interval conditions.
5) What does ω(δ) represent here?
ω(δ) is the sampled modulus of continuity. It measures the largest observed output change when two sampled inputs are within delta.
6) Why does the calculator estimate a Lipschitz bound?
A bounded slope suggests Lipschitz behavior. Then output differences scale with input differences, giving a practical sufficient delta estimate.
7) Why might tan(x) or sec(x) give issues?
These functions can become undefined near vertical asymptotes. If the interval includes such points, continuity and uniform continuity may fail.
8) How can I improve calculation accuracy?
Increase the sample count, increase delta steps, use a tighter interval, and avoid intervals containing undefined points or very sharp oscillations.