Continuity and the Intermediate Value Theorem
Define continuity at a point as the limit agreeing with the value there, classify the ways that can fail, and use the intermediate value theorem to guarantee a root inside an interval.
What a learner can do afterwards
- Check point and endpoint continuity conditions and determine needed point-value repairs
- Analyze removable, jump and infinite behavior, including parameter choices for matching branches
- Apply the Intermediate Value Theorem to roots or target values and distinguish interior guarantees from endpoint values
1 · Read
At an interior point c, continuity requires f(c) to be defined, a finite two-sided limit to exist, and equality between that limit and f(c). A derivative is not required. A corner can be continuous even when its derivative does not exist.
If a finite limit at c equals the defined value f(c), all continuity conditions hold. If the limit and value differ, continuity fails even though both exist. Changing the value to the finite limit repairs that mismatch. On a closed interval, endpoint continuity uses only the available side: right at the left endpoint, left at the right endpoint.
An infinite discontinuity has outputs tending without bound on at least one side, giving a vertical asymptote. For example, 1/(x+6)² grows toward positive infinity on both sides of -6. No assigned value there can turn this behavior into a finite limit.
A removable discontinuity has a finite limit but a missing or mismatched point value. For (x²-16)/(x-4), cancellation away from 4 leaves x+4. Its limit at 4 is 8, so defining its value there as 8 fills the hole. A different assigned value would leave a removable discontinuity.
A jump has unequal finite one-sided limits. For x+2 below 1 and 6 above 1, the limits are 3 and 6. No point assignment repairs this disagreement. With an adjustable branch parameter, first equate the one-sided limits. Matching them and assigning the shared value establishes continuity.
The Intermediate Value Theorem applies to a function continuous on [a,b]. Every target strictly between its endpoint outputs is attained in (a,b). Opposite nonzero endpoint signs therefore guarantee at least one interior root. Separate sign changes on disjoint open intervals guarantee distinct roots in those intervals. Polynomials are continuous on every real interval.
Polynomials are continuous. For x³-3 on [1,2], the endpoint values are -2 and 5. Zero lies strictly between them, so at least one root lies in (1,2). The theorem does not specify its exact input or prove a total root count. The same argument works for nonzero target values between endpoint outputs.
Continuity on the whole interval is essential. A function equal to -3 on one side of a jump and 4 on the other can skip zero despite opposite endpoint signs. Without continuity, the Intermediate Value Theorem supplies no root guarantee. This does not assert that every discontinuous function lacks roots.
Endpoint outputs are attained at the endpoints themselves. The theorem's interior guarantee is for strictly intermediate targets. An endpoint root does not establish another root inside. For example, f(x) = x on [0,2] has a root at 0 but none in (0,2). If both endpoints are roots, interior roots still need another argument.
Continuity matches the finite limit to the value. On a continuous interval, strictly intermediate outputs occur inside.
Original continuity and Intermediate Value Theorem lesson, questions and flashcards written for this learning dataset.
Practice questions written for this learning dataset. The rewritten readings were written for this dataset.
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