Test functions at specific points with structured checks. See continuity class, derivatives, and domain restrictions. Designed for study, teaching, revision, and quick verification tasks.
Choose a function model, enter coefficients, set the target point, and evaluate continuity and differentiability.
| Function family | Sample expression | Point | Continuity | Differentiability |
|---|---|---|---|---|
| Polynomial | x^3 - 2x + 1 | x = 1 | Continuous | Differentiable |
| Rational | (x + 1) / (x - 2) | x = 2 | Discontinuous | Not differentiable |
| Radical | sqrt(x) | x = 0 | Continuous on domain | Not differentiable |
| Absolute value | |x - 3| | x = 3 | Continuous | Not differentiable |
The calculator checks continuity by combining domain rules, point value checks, and nearby left and right samples. It checks differentiability with exact derivative formulas whenever the model allows them.
Continuity tells you whether a function behaves without a break at a chosen point. A function is continuous when the value exists, the limit exists, and both are equal. This idea matters in calculus, graph analysis, optimization, and motion problems. A continuity calculator helps students test these conditions quickly and clearly.
Differentiability goes one step further. It asks whether the function has a stable slope at that point. If the derivative exists, the graph has no sharp corner, cusp, or vertical tangent there. Every differentiable function is continuous at that point. The reverse is not always true. The classic example is an absolute value graph at its corner.
This calculator focuses on pointwise analysis. That approach is practical for exams and homework. You can test a polynomial, rational, radical, logarithmic, exponential, trigonometric, or absolute value model with one form. You also get domain guidance, exact derivative rules, and left and right numerical checks. Those details make the result easier to interpret.
Polynomial, sine, cosine, and exponential models are usually smooth on all real numbers. Rational functions break when the denominator becomes zero. Logarithmic functions only work when the argument is positive. Radical functions require a nonnegative radicand. Absolute value models stay continuous but may fail differentiability at a corner. These are standard patterns in real analysis and introductory calculus.
Students often confuse undefined points with nondifferentiable points. They also mix up removable discontinuities and infinite discontinuities. This calculator separates those cases with direct statements. It shows the function value, nearby samples, the continuity decision, and the derivative status in one report. That makes revision faster and helps build strong mathematical intuition for limits, domains, and slopes.
Yes. Absolute value at its corner is the classic example. The graph has no break, so it is continuous. The slope changes abruptly, so the derivative does not exist there.
Yes. Differentiability at a point guarantees continuity at that same point. If a function is not continuous, it cannot be differentiable there.
A rational function fails continuity when its denominator becomes zero. That creates an undefined point. The break may be removable or may behave like a vertical asymptote.
The function is defined and continuous on its domain at zero. However, the slope grows without bound near that boundary point, so the derivative does not exist there in the usual sense.
The step size controls how close the left and right sample points are to the chosen point. Smaller values usually give a better numerical picture of local behavior.
This version supports several common function families with coefficients. That keeps the analysis reliable and makes the continuity and derivative rules exact for each supported model.
A removable discontinuity happens when a function is undefined at a point, but the nearby limit still exists. Redefining the function value there can patch the hole.
They help detect corners, cusps, and sudden slope changes. When the left and right derivative estimates disagree, differentiability usually fails at that point.
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.