Compare the limit and the value
\(\begin{aligned}&\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)\\&\lim_{x\to a}f(x)=f(a)\end{aligned}\)
For a piecewise function, use the appropriate rule on each side and at the point.
Tahat Just Math · Self-paced practice
Understand and Practice Continuity
Check continuity at a point, classify discontinuities, and apply the Intermediate Value Theorem.
- Determine whether a function is continuous at a point.
- Classify removable and jump discontinuities.
- Determine continuity on intervals and apply the Intermediate Value Theorem.
15 questions · 3 learning objectives
At a point
Check all three conditions
A function is continuous at \(x=a\) when the value \(f(a)\) is defined, the two-sided limit exists, and the two are equal.
Discontinuities
Identify why continuity fails
Removable
The finite two-sided limit exists, but the function value is missing or different. Defining \(f(a)\) to equal that limit removes the discontinuity.
Jump
The finite left- and right-hand limits differ. Changing only \(f(a)\) cannot make them agree.
Infinite
At least one one-sided limit is \(+\infty\) or \(-\infty\), indicating a vertical asymptote.
Intervals
Check the whole interval
Continuity on a closed interval requires continuity inside it and the appropriate one-sided continuity at each endpoint. Rational functions are continuous wherever their denominators are nonzero.
Example
\(f(x)=1/(x-2)\) is continuous on \([-1,1]\), because \(x=2\) is outside the interval. It is not continuous on \([1,3]\), which contains \(2\).
Intermediate Value Theorem
Find a guaranteed value
If \(f\) is continuous on \([a,b]\), it takes every value between \(f(a)\) and \(f(b)\). A value strictly between the endpoint values occurs at some \(c\) in \((a,b)\); the theorem does not identify \(c\) or guarantee uniqueness.
Guaranteeing a zero
For \(f(x)=x^3+x-1\), the function is continuous on \([0,1]\), with \(f(0)=-1\) and \(f(1)=1\). Therefore, at least one \(c\in(0,1)\) satisfies \(f(c)=0\).
Continuity Lab
Choose an example, then compare its one-sided limits and function value at the joining point.
Learn by doing
Guided examples by skill
Open a skill, read the idea, then work through one example a step at a time. Use each question’s hint and step-by-step solution controls in Practice problems as needed.
Chapter 2 · Mock exam
A focused 20-question test
Questions are selected at random from the objectives you choose, with at least one question from each selected objective. Answer in any order, flag questions to revisit, and review everything before you submit. Your score and answer review appear at the end. Use your school-approved tablet; phones should remain put away under classroom rules.
- 20random questions
- 40 minutesauto-submits when time ends
- Tab & focus monitoringends after 3 departures
Switching tabs or focusing another window is counted. Some split-screen modes keep both pages focused and visible, which a website cannot detect. This is not secure proctoring.
The test requests fullscreen when you start. Exiting fullscreen is recorded and pauses the test until you return to fullscreen. If fullscreen is unavailable, you can choose to continue without it.
Focused practice
Choose a learning objective
Practice one skill at a time, or use the full assessment from the hero section to review the entire chapter.
Random practice
Mixed practice across every objective
Chapter 2 contains 15 questions across 3 learning objectives. Choose any number of questions you like. A random set takes one question from each objective first, so a set of 8 or more questions covers the whole chapter.