Discontinuity Finder Calculator

Calculator Inputs

Function family

Range start

Range end

a coefficient

b coefficient

c constant / phase

Numerator x coefficient

Numerator constant

Cancellation check

Radical denominator shift

Radical model

Log base

Rational uses (numA·x + numB) / (a·x + b). Logarithmic uses a·log_base(b·x + c). Radical checks √(a·x + b)/(x + shift) or 1/√(a·x + b). Tangent uses a·tan(b·x + c).

Example Data Table

Function family	Sample inputs	Expected discontinuity	Type
Rational	(x - 2) / (x - 2)	x = 2	Removable hole
Rational	(x + 1) / (x - 4)	x = 4	Vertical asymptote
Logarithmic	log(3x - 6)	x = 2 boundary	Domain break
Radical	1 / √(x - 9)	x = 9	Non-permitted point
Tangent	tan(x)	x = π/2 + kπ	Periodic asymptotes

Formula Used

Rational: discontinuities occur where the denominator equals zero. If the same factor appears in the numerator, the point becomes removable; otherwise, it is a vertical asymptote.

Logarithmic: for log(bx + c), the expression inside the log must stay positive. The boundary bx + c = 0 marks a domain discontinuity.

Radical: for √(ax + b), the radicand must be at least zero. For 1/√(ax + b), it must be strictly greater than zero.

Tangent: tan(θ) is undefined when cos(θ) = 0. Thus, asymptotes occur at θ = π/2 + kπ, where k is any integer.

How to Use This Calculator

Select the function family that matches your expression.
Enter the coefficients, constants, and visible x-range.
Press Find Discontinuities to generate the report.
Review the summary, domain notes, and listed breakpoints.
Download the result as CSV or PDF if needed.

FAQs

What does a discontinuity mean?

It is a point where a function breaks, becomes undefined, jumps, or approaches infinity. The graph cannot be traced smoothly through that location.

What is a removable discontinuity?

A removable discontinuity is a hole caused by a shared factor that cancels. The function is undefined at one point, but nearby values still follow one limit.

How is a vertical asymptote different?

A vertical asymptote appears when the function grows without bound near a restricted x-value. Unlike a hole, the nearby behavior becomes unbounded.

Why do logarithms create domain breaks?

Logarithms need positive inputs. When the inside expression becomes zero or negative, the function is undefined and the domain breaks there.

Why is tangent discontinuous periodically?

Tangent equals sine divided by cosine. Whenever cosine becomes zero, tangent is undefined, creating repeating vertical asymptotes across the real line.

Can the calculator prove continuity everywhere else?

It identifies standard discontinuity rules for the supported families and the chosen range. It does not replace a full symbolic proof for all expressions.

Why choose a visible x-range?

Some families, especially tangent, have infinitely many discontinuities. The range lets the tool list only the breakpoints relevant to your interval.