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Function Continuity Calculator

Check whether a function is continuous at x = a and classify any discontinuity: removable, jump, infinite, or oscillatory. Includes a table, a mini graph preview, and a solve-for-k helper to “make it continuous”.

Background

A function is continuous at x=a if: (1) f(a) exists, (2) limx→a f(x) exists, and (3) they’re equal.

Enter inputs

Tip: “Solve for k” is for piecewise definitions like f(a)=k.

Use x as the variable. Use ^ for powers (like x^2).

Options

Chips fill values and calculate immediately.

Result:

No results yet — enter inputs and click Calculate.

How to use this calculator

  • Continuity at a point: enter f(x) and the point a.
  • We compute left limit, right limit, two-sided limit, and f(a).
  • Solve for k: enter g(x) (for x≠a) and we solve k=limx→ag(x).
  • Piecewise boundary: enter left and right rules around a and pick which side defines f(a).
  • Optional: turn on differentiability to detect corners and cusps at a boundary.

How this calculator works

  • Estimates limits using shrinking steps toward a from the left and right.
  • Classifies behavior as finite, jump, infinite, or oscillatory.
  • Checks continuity using: limit exists + f(a) exists + equality.
  • If enabled, estimates one-sided derivatives using one-sided difference quotients.

Formula & Equation Used

Continuity at a point: f is continuous at a if lim exists and equals f(a).

lim x→a f(x) = f(a)

Two-sided limit exists when left and right limits match:

limx→a− f(x) = limx→a+ f(x)

One-sided derivatives (used in the differentiability check):

f′(a−) = lim h→0+ f(a)−f(a−h) h

f′(a+) = lim h→0+ f(a+h)−f(a) h

Example Problem & Step-by-Step Solution

Example 1 — Removable discontinuity

Determine continuity at a=1 for (x^2 − 1)/(x − 1).

  • Direct substitution gives 0/0 (undefined), so f(1) is missing.
  • But the simplified rule is x+1 (for x≠1), so the limit is 2.
  • Limit exists, but f(1) is undefined → removable discontinuity.

Example 2 — Jump discontinuity

Check continuity at a=0 for |x|/x.

  • As x→0−, |x|/x = −1.
  • As x→0+, |x|/x = 1.
  • Left ≠ right → two-sided limit DNE → jump discontinuity.

Example 3 — Infinite discontinuity

Check continuity at a=2 for 1/(x−2).

  • As x→2, values blow up to ±∞.
  • That’s a vertical asymptote → infinite discontinuity.

Example 4 — Boundary corner (differentiability)

Let f(x)=−x for x≤0 and f(x)=x for x≥0. At a=0, the function is continuous but not differentiable.

  • Left limit equals right limit equals 0 → continuous.
  • Left derivative ≈ −1, right derivative ≈ 1.
  • Finite but unequal → corner.

Frequently Asked Questions

Q: Can a function be continuous even if f(a) looks “weird”?

Yes. Continuity only cares whether the limit exists and equals f(a). The function can be complicated but still continuous.

Q: Can the limit exist if f(a) is undefined?

Yes. That’s the classic removable discontinuity situation: the function approaches a value, but the point is missing (a “hole”).

Q: What’s the difference between removable and jump discontinuities?

Removable: left and right match (limit exists). Jump: left and right are different (limit does not exist).

Q: If a function is continuous, is it automatically differentiable?

No. Corners/cusps can be continuous but not differentiable (like |x| at 0).

Q: How should I use the table and graph?

They’re sanity checks. If the left side and right side behave differently, you’ll see it immediately.