Continuity Checker

Math Calculus • Limits and Continuity

Written by STEM Calculators Team Published December 21, 2025 Updated February 24, 2026

3. Continuity Checker

Test whether \(f(x)\) is continuous at a point \(x=a\) using the three conditions: \(f(a)\) is defined, \(\lim_{x\to a} f(x)\) exists, and \(\lim_{x\to a} f(x)=f(a)\). Classifies removable, jump, and infinite discontinuities (with “fix” suggestions).

Inputs

Function \(f(x)\)

Supported: + − * / ^, parentheses, x, pi, e, sin cos tan, ln log (base 10), sqrt abs exp. Implicit multiplication is allowed: 2x, (x+1)(x-1), 2sin(x).
Piecewise: piecewise(expr1, cond1, expr2, cond2, ..., defaultExpr) where a condition can be like x<1, x>=0, x=2, x!=3.

Point \(a\)

Accepts constants like pi, e, and expressions like 2*pi.

Check mode

Interval scan is a best-effort numeric detector (useful for spotting breaks).

Approach depth

Uses \(a\pm 10^{-k}\), \(k=1..N\).

Graph window (half-width)

Graph uses \(x\in[a-w,a+w]\).

Interval \(L\)

Used only in interval scan mode.

Interval \(R\)

Used only in interval scan mode.

Options Try rational (polynomial) simplification Animate approach (number line + points) Show discontinuity markers/labels

Ready

Graph and highlight

Drag to pan • wheel/pinch to zoom • dashed line is \(x=a\) • open/filled points highlight discontinuities

Animated approach on a number line:

x: 0, y: 0, zoom: 1

Result

Enter a function and click Check.

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3. Continuity Checker — Theory

This tool checks continuity at a point \(x=a\) using the standard three-condition test and classifies common discontinuities.

1) What it means to be continuous at \(x=a\)

A function \(f\) is continuous at \(x=a\) if and only if all three conditions hold:

\[ \text{(1) } f(a)\ \text{is defined} \quad \text{(2) } \lim_{x\to a} f(x)\ \text{exists (finite)} \quad \text{(3) } \lim_{x\to a} f(x) = f(a). \]

If any condition fails, \(f\) is discontinuous at \(a\).

2) One-sided limits

The two-sided limit exists when the one-sided limits agree:

\[ \lim_{x\to a} f(x)\ \text{exists} \iff \lim_{x\to a^-} f(x) = \lim_{x\to a^+} f(x). \]

If the left-hand and right-hand limits are different, the two-sided limit is DNE (does not exist).

3) Types of discontinuities

Removable discontinuity (a “hole”)

The limit exists and is finite, but \(f(a)\) is missing or wrong:

\[ \lim_{x\to a} f(x)=L\ \text{(finite)}, \quad \text{but } f(a)\ \text{is undefined or } f(a)\ne L. \]

You can “fix” it by redefining \(f(a)=L\).

\[ f(x)=\frac{x^2-1}{x-1}=\frac{(x-1)(x+1)}{x-1}=x+1 \quad (x\ne 1) \]

Here \(f(1)\) is undefined, but \(\lim_{x\to 1} f(x)=2\). Defining \(f(1)=2\) makes it continuous.

Jump discontinuity

Both one-sided limits exist (finite) but they are different:

\[ \lim_{x\to a^-} f(x) = L_-,\quad \lim_{x\to a^+} f(x) = L_+,\quad L_- \ne L_+. \]

A typical cause is a piecewise-defined function with a “step”.

Infinite discontinuity (vertical asymptote)

At least one one-sided limit diverges to \(\pm\infty\):

\[ \lim_{x\to a^-} f(x)=\pm\infty \quad \text{or} \quad \lim_{x\to a^+} f(x)=\pm\infty. \]

Example: \(f(x)=\frac{1}{x-2}\) has an infinite discontinuity at \(x=2\).

4) Continuity on an interval

A function is continuous on an interval if it is continuous at every point in the interval. The calculator’s interval mode performs a numeric scan to highlight candidate breakpoints, then you can test each candidate precisely in point mode.

Frequently Asked Questions

How do I check if a function is continuous at x=a?

Continuity at x=a requires three conditions: f(a) is defined, lim_{x->a} f(x) exists, and lim_{x->a} f(x)=f(a). Enter f(x) and a, then run the checker to see whether each condition holds.

What is a removable discontinuity and how can it be fixed?

A removable discontinuity occurs when the limit lim_{x->a} f(x) exists and is finite, but f(a) is missing or not equal to that limit. It can often be fixed by redefining f(a) to equal the limit value.

What is the difference between a jump and an infinite discontinuity?

A jump discontinuity happens when the left-hand and right-hand limits both exist and are finite but are not equal. An infinite discontinuity happens when at least one one-sided limit diverges to infinity or -infinity, indicating a vertical asymptote.

How does the interval scan mode find discontinuities?

Interval scan mode performs a best-effort numeric sweep over [L, R] to flag likely breakpoints where the function behavior changes sharply. The flagged points are candidates that you can test more precisely in point mode.