Calculus learning

Continuity and Discontinuities: Values, Limits, Holes and Jumps

Continuity is a three-part check, not just the existence of a limit. A removable hole can have a finite limit but no function value. A jump has incompatible one-sided limits; an infinite discontinuity has unbounded behavior near the point.

Direct answer

At an interior point a, continuity requires f(a) to exist, lim x→a f(x) to exist, and the two to agree.

At a domain endpoint use the appropriate one-sided condition. Continuity and differentiability are different properties.

Continuity and Discontinuities worked method

For (x²−4)/(x−2), cancel to x+2 only when x≠2. The limit at 2 is 4, but the original function is undefined there.

Defining the missing value as 4 removes that hole. Defining it as any other number leaves the function discontinuous.

A piecewise function that equals −1 for x<0 and 1 for x≥0 jumps at zero: the left limit is −1 and the right limit is 1.

For 1/x at zero, the left and right behaviors are unbounded with opposite signs. No single finite value repairs the discontinuity.

The function |x| is continuous at zero, but its one-sided slopes disagree. A continuous graph can therefore fail to have a derivative.

1
lim x→2 (x²−4)/(x−2)=4, but f(2) is undefined
2
Continuity: f(a)=lim x→a f(x)
3
|x| is continuous at 0, not differentiable at 0

Worked examples

(x²−4)/(x−2) at 2
Removable hole

Set f(2)=4 to repair it.

1/x at 0
Infinite discontinuity

The two sides do not have the same finite limit.

|x| at 0
Continuous, not differentiable

Continuity does not guarantee a tangent slope.

Common mistakes

  • • Restoring excluded points after cancellation without defining them.
  • • Checking only the right limit at an interior point.
  • • Equating continuity with differentiability.

Try the supported calculator preset

The preset opens a supported expression with its variable and conditions. Teaching examples elsewhere in this guide are reference explanations, not claims that the engine supports every method.

The preset computes the limit, not f(2). The lesson preserves the original exclusion and does not call the function continuous.

Limits:x2−4x−2\frac{x^{2} - 4}{x - 2}
Open the calculator without a preset: Limits

References

Frequently asked questions

Can a function be discontinuous where its limit exists?+

Yes, if its value is missing or differs from that limit.

Can I repair a jump by changing one point?+

No. Changing one value does not reconcile different one-sided limits.

Does the preset classify arbitrary piecewise functions?+

No. It evaluates a rational-hole limit; the complete continuity check also needs the original point value.

Conclusion

Continuity and Discontinuities starts with the stated domain and conditions. Recheck those before carrying a worked example into a different problem.

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