Calculus I · Limits and Continuity · quiz

Continuity Concept Quiz

Section 5 Concept Quiz

Use this as a short retrieval check after finishing the section. On the website, each item should accept an answer before revealing the hint and worked explanation.

Interactive checkcontinuity-q1

Is $f(x)=x^2+1$ continuous at $x=3$ ? Enter yes or no.

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Polynomials are continuous everywhere.

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Interactive checkrepair-q1

Choose $c$ so $f(x)=\frac{x^2-9}{x-3}$ for $x\ne3$ , and $f(3)=c$ , is continuous.

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Factor, cancel, and find the approached value at $3$ .

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Interactive checkpiecewise-q1

Choose $a$ so $2x+a$ for $x<1$ joins continuously to $x^2+3$ for $x\ge1$ .

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Set the left value at $1$ equal to the right value at $1$ .

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Interactive checkivt-q1

Does the IVT guarantee a zero of $x^3+x-1$ on $[0,1]$ ? Enter yes or no.

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Check continuity and the signs at both endpoints.

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Summary

Quiz use on the website

This page should report completion by concept, not merely a total score. A wrong answer should link back to the exact lesson that teaches the missing method. The same questions may appear in randomized numerical variants, but the canonical explanation should remain stable.

After the explanation

Use the section idea

Reading lens

Do the limit, the function value, and the surrounding domain fit together at the point or across the interval?

Mental model

Continuity is a three-part agreement: the value exists, the two-sided limit exists, and those two quantities are equal.

Decision

At a point, test the three conditions in order; on an interval, check the domain and endpoints before invoking any continuity theorem.

Common trap

A sign change supports the Intermediate Value Theorem only when continuity holds on the entire closed interval, and it does not prove uniqueness.

Check yourself

You are ready to continue when you can classify a break, decide whether one value can repair it, and state every IVT hypothesis aloud.

Source & rights

Original instruction with traceable references.

The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.

The verified handoff declares original composition and requires owner provenance review. BetterGrades-original material remains separate from public-domain references; no source textbook PDF is published here.