Enter Function Details
Formula Used
Continuity at x = c:
f(c) exists, lim x→c f(x) exists, and lim x→c f(x) = f(c).
One-sided limit check:
L⁻ = lim x→c⁻ f(x) and L⁺ = lim x→c⁺ f(x).
Classification:
If L⁻ = L⁺ = f(c), the function is continuous.
If L⁻ = L⁺ but f(c) is missing or different, it is removable.
If L⁻ ≠ L⁺, it is a jump discontinuity.
If values grow without bound, it is an infinite discontinuity.
How to Use This Calculator
- Select single expression or piecewise mode.
- Enter the function using
xas the variable. - Use operators such as
+,-,*,/, and^. - Use functions like
sin(x),cos(x),sqrt(x),log(x),abs(x), andpow(x,2). - Enter the point where continuity should be tested.
- Adjust tolerance and step size when the answer needs finer inspection.
- Press the button to see limits, classification, graph, and table.
- Use CSV or PDF export to save the result.
Example Data Table
| Function | Test point | Expected behavior | Type |
|---|---|---|---|
(x^2 - 1) / (x - 1) |
1 | Limit is 2, but value is undefined. | Removable discontinuity |
abs(x) |
0 | Left limit, right limit, and value match. | Continuous |
1 / (x - 2) |
2 | Values grow without bound near the point. | Infinite discontinuity |
Piecewise: left -1, right 1 |
0 | Side limits approach different values. | Jump discontinuity |
Understanding Continuous and Discontinuous Functions
Understanding Continuous Graphs
A continuous graph can be traced without lifting the pencil. This common idea is helpful. Yet the formal test is stricter. A function is continuous at a point when three facts agree. The function value exists. The left limit and right limit exist. Both side limits equal the function value. When this happens, the graph behaves smoothly at that point.
Why Discontinuities Matter
A discontinuity shows a break in the rule or graph. It can be removable, like a missing point. It can be a jump, where each side approaches a different height. It can be infinite, where values grow without bound near the point. Some functions also oscillate near a point. Those cases need careful review. This calculator focuses on numerical side checks and graph inspection.
How This Calculator Helps
Enter a function and the point you want to test. The tool samples values near the point from the left and right. It compares the estimated one-sided limits with the defined function value. The answer includes a classification, a difference report, and a plot. The plot helps you see holes, jumps, and sharp changes. The table gives nearby values for audit checks. Export buttons let you save the calculation.
Best Use Tips
Use smaller steps for smooth functions. Use a wider graph range for large features. Use tighter tolerance only when inputs are clean. Numerical tools can be affected by rounding. They also struggle with severe oscillation. Always compare the formula, the table, and the chart. This is important for piecewise rules. For classroom work, write the limit statement after reviewing the result. For applied work, treat jumps as decision points. A discontinuity can mark a threshold, boundary, or model failure. Clear continuity checks make graph behavior easier to explain.
Common Mistakes to Avoid
Do not judge continuity from a picture alone. A graph may look connected at low zoom. The formula may still be undefined. Also check the exact point. Nearby values do not replace the actual value. If a denominator becomes zero, review possible cancellation. If a piecewise rule changes at the point, test each branch separately. Use notes to explain each final conclusion.
FAQs
What is a continuous graph?
A continuous graph has no break at the tested point. The left limit, right limit, and actual function value all match. The graph can pass through that point smoothly.
What is a discontinuous function?
A discontinuous function fails the continuity test at one or more points. The function may have a hole, jump, vertical break, undefined value, or unstable behavior.
What is a removable discontinuity?
A removable discontinuity happens when both side limits agree, but the function value is missing or different. It often appears as a hole in the graph.
What is a jump discontinuity?
A jump discontinuity occurs when the left limit and right limit are finite but different. The graph jumps from one height to another near the point.
What is an infinite discontinuity?
An infinite discontinuity appears when values grow very large near the point. This often means the graph has a vertical asymptote or division by zero behavior.
Can this calculator prove continuity exactly?
It gives a strong numerical check and graph view. For exact proof, simplify the function, check the domain, and write formal limit statements.
Which functions can I enter?
You can use common functions like sin, cos, tan, sqrt, abs, log, exp, pow, min, and max. Use x as the variable.
Why should I adjust tolerance?
Tolerance controls how closely values must match. Smaller tolerance is stricter. Larger tolerance can help when rounding or numerical noise affects nearby values.