IP331.com | Online Tools

Algebra Tools Geometry Tools Trigonometry Tools Matrix Tools Function Tools Number Theory Tools Base Converter Electronics & Physics Statistics Tools Life & Finance

Home›Function Tools›Continuity Tester

Continuity Tester

Check if a function is continuous at a point or over its domain

Select Function Type

Polynomial Rational Piecewise At x=a

Polynomial: ax^n + bx^(n-1) + ...

f(x)=x^

Continuity Definition

f continuous at a if: lim_{x->a} f(x) = f(a)

1) f(a) defined 2) lim exists 3) lim = f(a)

Polynomials: continuous everywhere

Rational: continuous except denominator=0

Continuity means the graph can be drawn without lifting the pen. A function is continuous at a point if the function value, left limit, and right limit all agree. Polynomials are always continuous; rational functions have discontinuities at denominator zeros.

⚠ Three conditions for continuity at a: f(a) defined, limit exists, limit = f(a). Break any one and you have a discontinuity.

What Is Continuity?

Continuity is a fundamental concept in calculus. A continuous function has no breaks, jumps, or holes. Polynomials are continuous on R. Rationals have discontinuities at points where denominator=0 (vertical asymptotes or holes if numerator also zero).

Polynomials

Always continuous on R. No breaks, holes, or asymptotes. lim x->a P(x) = P(a) for all a.

Rational Functions

Continuous except where denominator=0. Vertical asymptote if numerator != 0. Hole if both are zero.

Piecewise

Check continuity at the boundary point. Compare left limit, right limit, and function value. Must all match.

Discontinuity Types

Removable (hole): limit exists. Jump: left!=right limit. Infinite: limit = +/-inf. Essential: oscillation.

Teaching Example: f(x)=1/(x-2) at x=2. f(2) undefined. Left limit-> -inf, right limit-> +inf. Discontinuous (infinite discontinuity/vertical asymptote).

Applications

Calculus Limit Analysis Differentiability Integration Graphing

Frequently Asked Questions

What is continuity?▼

lim f(x) = f(a). Three checks: defined at a, limit exists, limit equals value. Unbroken graph.

Discontinuity types?▼

Removable (hole), jump (different sides), infinite (asymptote), essential (oscillation).

Are polynomials continuous?▼

Yes, all polynomials are continuous on R. No breaks or holes. The limit always equals the function value.

When does rational become discontinuous?▼

At denominator zeros. If numerator also zero, it is a hole (removable). Otherwise, a vertical asymptote (infinite discontinuity).

More Function Tools

Strictly Concave Analyzer

Verify if f(x) < 0 (strictly concave) for common functions

›

Function Monotonicity Calculator

Analyze increasing/decreasing behavior and find monotonic intervals

›

Function Growth Rate Analyzer

Compare asymptotic growth rates of functions using limits

›

Absolute Extrema Finder

Find global min and max of functions on closed intervals

›

Relative Extrema Finder

Find local minima and maxima using derivative tests

›

Exponential Function Solver

Solve a^x=b and evaluate exponential functions using logarithms

›