Chapter 4 · Test 1
Continuity for Economics
Limits of functions, continuity tests, and economic discontinuities.
What you will learn
- Choose a parameter for continuity
- Evaluate a removable limit
- Classify a discontinuity
Worked example
Choose a parameter for continuity: follow the method step by step
Try the problem first, then compare your approach with the complete solution and the reasoning behind it.
Choose a parameter for continuity
Let f(x) = kx + 1 for x < 2 and f(x) = 7 for x ≥ 2. What value of k makes f continuous at x = 2?
Continuity at x = 2 requires limₓ→₂⁻ f(x) = f(2)
In plain English: For the function to be continuous at 2, its value as x approaches 2 from the left must equal its actual value at 2.
Bold blue terms in this symbol guide include definitions. Hover, focus, or tap a term. Each row also shows how to read the notation aloud.
- x
- the input variableHow to read: x
- k
- the Parameter: A value treated as fixed while a model is solved, although it may be changed to compare different scenarios. to determineHow to read: k
- f(x)
- the piecewise functionHow to read: f of x
Solution
- From the right-hand branch, f(2) = 7.
- The left-hand limit is 2k + 1.
- Impose continuity: 2k + 1 = 7.
- Subtract 1: 2k = 6.
- Divide by 2: k = 3.
Visual check
Piecewise function at the join
An open circle excludes the left-piece endpoint. A filled point includes the value supplied by the right piece.
- Left piece
- Right piece
How to read this graph
- Horizontal axis: x
- Vertical axis: f(x)
- Curves and lines: Left piece, Right piece
- Marked points: Open the exact-coordinate table below to read their values.
What the graph shows: The left-hand value approaches 7 at x = 2, while the function value from the right piece is 7.
Read the marked points as data
| Feature | x | f(x) |
|---|---|---|
| Left-hand limit 7 | 2 | 7 |
| f(2) = 7 | 2 | 7 |
Logic and solving tips
- At a joining point, continuity requires the two branch values and the function value to agree.
- At the join, take each branch expression's one-sided limit and set those limiting values equal.
Where economists use it
A policy designer chooses the connecting parameter in a piecewise tax or benefit schedule so earning one additional dollar does not create an unintended jump in disposable income.
Key idea: At a joining point, continuity requires the two branch values and the function value to agree.
Applied case study
Joining the EITC phase-in to its maximum credit
Scenario
Policy analysts impose continuity when joining pieces of tax and benefit schedules. IRS lists a $13,020 earned-income amount and $4,427 maximum 2026 EITC for a taxpayer with one qualifying child.
Problem
Use the compact schedule E(y) = 0.34y for y < 13,020 and E(y) = k at the start of the plateau. Choose k so the schedule is continuous.
lim[y→13,020⁻] E(y) = E(13,020) 0.34(13,020) = k
In plain English: The limit asks for the value the expression approaches as its input moves arbitrarily close to the stated point.
Worked solution
- Evaluate the phase-in branch at the boundary: 0.34(13,020).
- Multiply: 0.34(13,020) = 4,426.80.
- Set the plateau equal to the left-hand limit: k = 4,426.80.
- In whole-dollar tax-table terms, k ≈ $4,427, matching the published maximum.
Economic interpretation: Equal branch values avoid a benefit jump where the phase-in reaches the maximum credit.
Source: Internal Revenue Service: Internal Revenue Bulletin 2025-45 The 2026 boundary and maximum are sourced. The compact function omits filing status, phaseout, rounding, and other eligibility rules and is only a teaching model.
Reasoning habits that help
- At a joining point, continuity requires the two branch values and the function value to agree.
- A hole at one point does not prevent a limit from existing when nearby values approach a single number.
- Continuity classification compares the side limits and function value to identify smooth, removable, jump, or unbounded behavior.
Where economists use these methods
- A policy designer chooses the connecting parameter in a piecewise tax or benefit schedule so earning one additional dollar does not create an unintended jump in disposable income.
- An economist evaluates a limit when average cost, tax burden, or another ratio is undefined at one point but its nearby behavior still determines the economically relevant value.
- A public-finance analyst checks a benefit threshold for a sudden income loss, distinguishing a genuine policy cliff from a removable formula error or an unbounded rule.
Chapter vocabulary
Study 17 key terms
Say what the term means before opening it. Then compare your explanation with the definition, notation, and example.
Pointwise continuity
Continuity evaluated at one specified input value.
Textbook reference: Chapter 4, Section 4.1 Continuity of a Function of One Variable, printed page 103Left-hand limitlim_{x→a^-} f(x)
The value a function approaches as its input approaches a point through smaller values only.
How to read: the limit of f of x as x approaches a from the left
Textbook reference: Chapter 4, Section 4.1 Continuity of a Function of One Variable, printed page 104Right-hand limitlim_{x→a^+} f(x)
The value a function approaches as its input approaches a point through larger values only.
How to read: the limit of f of x as x approaches a from the right
Textbook reference: Chapter 4, Section 4.1 Continuity of a Function of One Variable, printed pages 104-105Two-sided limitlim_{x→a} f(x) = L
A limit that exists when the left-hand and right-hand limits both exist and are equal.
Textbook reference: Chapter 4, Section 4.1 Continuity of a Function of One Variable, printed page 105Continuity at a pointlim_{x→a} f(x) = f(a)
A function is continuous at a when f(a) exists, the limit as x approaches a exists, and that limit equals f(a).
Textbook reference: Chapter 4, Section 4.1 Continuity of a Function of One Variable, printed pages 105-106Continuous function
A function that is continuous at every point in its domain.
Textbook reference: Chapter 4, Section 4.1 Continuity of a Function of One Variable, printed page 106Discontinuous function
A function that fails at least one continuity requirement at one or more points in its domain.
Textbook reference: Chapter 4, Section 4.1 Continuity of a Function of One Variable, printed pages 106-107Jump discontinuity
A discontinuity where finite left-hand and right-hand limits exist but are not equal.
Textbook reference: Chapter 4, Section 4.1 Continuity of a Function of One Variable, printed page 107Removable discontinuity
A discontinuity where a finite two-sided limit exists but the function is missing or has a different value at the point.
Textbook reference: Chapter 4, Section 4.1 Continuity of a Function of One Variable, printed pages 107, 109Asymptote
A line that a function approaches arbitrarily closely in a limiting direction.
Textbook reference: Chapter 4, Section 4.1 Continuity of a Function of One Variable, printed pages 108-109Vertical asymptotex = a
A vertical line x = a near which a function becomes unbounded from at least one side.
Textbook reference: Chapter 4, Section 4.1 Continuity of a Function of One Variable, printed pages 108-109Continuous from the rightlim_{x→a^+} f(x) = f(a)
At a left endpoint a, the right-hand limit exists and equals f(a).
Textbook reference: Chapter 4, Section 4.1 Continuity of a Function of One Variable, printed page 111Continuous from the leftlim_{x→b^-} f(x) = f(b)
At a right endpoint b, the left-hand limit exists and equals f(b).
Textbook reference: Chapter 4, Section 4.1 Continuity of a Function of One Variable, printed page 111Continuity on a closed interval
Continuity at every interior point, from the right at the left endpoint, and from the left at the right endpoint.
Textbook reference: Chapter 4, Section 4.1 Continuity of a Function of One Variable, printed pages 110-111Piecewise-defined function
A function specified by different formulas on different parts of its domain.
Example: A tax schedule can apply one marginal rate below an income threshold and another above it.
Textbook reference: Chapter 4, Section 4.2 Economic Applications, printed page 113Marginal productMP(x) = Δq/Δx, or dq/dx when differentiable
The additional output associated with a small increase in an input, holding other inputs fixed.
Textbook reference: Chapter 4, Section 4.2 Economic Applications, printed page 116Bertrand price competition
A model in which firms choose prices and consumers buy from the lower-priced firm, creating payoff changes when prices cross or tie.
Textbook reference: Chapter 4, Section 4.2 Economic Applications, printed page 118End-of-chapter problems
Practice all 6 Chapter Review Exercises
The index covers every numbered Review Exercise in this chapter of Mathematics for Economics, 3rd edition. Each reference opens the textbook exercise itself, including every listed subpart, with MFE hints, a symbol guide, a worked explanation, progress tracking, and similar-problem practice.
Practice end-of-chapter problems