Unit 1.12 – Confirming Continuity over an Interval
AP® Calculus AB & BC | Formula Reference Sheet
Core Concept: Moving beyond checking continuity at a single point, this unit extends the concept to entire intervals. A function is continuous on an interval if it's continuous at every point in that interval. Think of it as ensuring the pencil never lifts while drawing the graph throughout the entire range—no breaks, no jumps, no holes anywhere!
📚 The Formal Definition
CONTINUITY ON AN INTERVAL
A function \(f(x)\) is continuous on an interval if it is continuous at every point in that interval.
College Board CED Statement: "A function is continuous on an interval if the function is continuous at each point in the interval."
📝 Key Insight: This definition means you must verify continuity requirements at every single point within the interval, not just a few sample points. However, for common function families (polynomials, trig, exponential, etc.), you can cite general continuity properties!
🎯 Types of Intervals
1. Open Interval: \((a, b)\)
Notation: \((a, b) = \{x : a < x < b\}\)
Meaning: Includes all numbers between \(a\) and \(b\), but NOT the endpoints themselves
Continuity requirement: Function must be continuous at every point strictly between \(a\) and \(b\)
Endpoints: Don't need to check \(x = a\) or \(x = b\) (they're not in the interval!)
2. Closed Interval: \([a, b]\)
Notation: \([a, b] = \{x : a \leq x \leq b\}\)
Meaning: Includes all numbers between \(a\) and \(b\), INCLUDING both endpoints
Continuity requirement:
- Continuous at all interior points: \(a < c < b\)
- Right-continuous at \(x = a\): \(\lim_{x \to a^+} f(x) = f(a)\)
- Left-continuous at \(x = b\): \(\lim_{x \to b^-} f(x) = f(b)\)
3. Half-Open Intervals: \([a, b)\) or \((a, b]\)
\([a, b)\): Includes \(a\) but NOT \(b\) → Check right-continuity at \(a\) only
\((a, b]\): Includes \(b\) but NOT \(a\) → Check left-continuity at \(b\) only
4. Infinite Intervals: \((a, \infty)\), \((-\infty, b)\), \((-\infty, \infty)\)
Meaning: Extends indefinitely in one or both directions
Continuity requirement: Continuous at all points within the interval; check one-sided limits at finite endpoints if included
⚠️ Critical Distinction: For open intervals, you only check interior points. For closed intervals, you MUST also verify one-sided continuity at the endpoints!
📏 Special Requirements for Closed Intervals \([a, b]\)
A function \(f\) is continuous on \([a, b]\) if:
- Interior Continuity: \(f\) is continuous at every \(c\) where \(a < c < b\)
\[ \lim_{x \to c} f(x) = f(c) \quad \text{for all } a < c < b \]
- Right-Continuity at Left Endpoint: \(f\) is continuous from the right at \(a\)
\[ \lim_{x \to a^+} f(x) = f(a) \]
- Left-Continuity at Right Endpoint: \(f\) is continuous from the left at \(b\)
\[ \lim_{x \to b^-} f(x) = f(b) \]
💡 Why One-Sided Limits at Endpoints? At the left endpoint \(a\), we can only approach from the right (since there are no points in the interval to the left of \(a\)). Similarly, at the right endpoint \(b\), we can only approach from the left. This is why we use one-sided limits at the boundaries!
✅ The Step-by-Step Strategy
How to Confirm Continuity on an Interval:
- STEP 1: Identify the interval type
- Open \((a, b)\), closed \([a, b]\), half-open, or infinite?
- This determines whether you need to check endpoints
- STEP 2: Check the domain
- Is the function even defined at all points in the interval?
- Look for: denominators = 0, negative under even roots, logs of non-positive, etc.
- If undefined anywhere in the interval → NOT continuous there
- STEP 3: Check interior points
- For "nice" functions (polynomials, trig, exponential, log on domain), cite continuity
- For piecewise or complex functions, check points where formula changes
- Look for holes, jumps, or asymptotes
- STEP 4: Check endpoints (if closed interval)
- At \(x = a\): Verify \(\lim_{x \to a^+} f(x) = f(a)\)
- At \(x = b\): Verify \(\lim_{x \to b^-} f(x) = f(b)\)
- STEP 5: State your conclusion
- If all checks pass: "Therefore, \(f\) is continuous on [interval]"
- If any check fails: "\(f\) is NOT continuous on [interval] because [reason]"
⭐ Functions Always Continuous on Their Domains
College Board Statement: "Polynomial, rational, power, exponential, logarithmic, and trigonometric functions are continuous on all points in their domains."
| Function Type | Domain | Continuity |
|---|---|---|
| Polynomials \(p(x) = a_nx^n + \cdots + a_0\) |
All real numbers \((-\infty, \infty)\) |
Continuous everywhere |
| Rational Functions \(\frac{p(x)}{q(x)}\) |
Where \(q(x) \neq 0\) | Continuous on domain |
| Exponential \(a^x, e^x\) |
All real numbers | Continuous everywhere |
| Logarithmic \(\ln(x), \log_a(x)\) |
\(x > 0\) | Continuous on \((0, \infty)\) |
| Trigonometric \(\sin(x), \cos(x)\) |
All real numbers | Continuous everywhere |
| \(\tan(x), \sec(x)\) | Where defined (not at \(\frac{\pi}{2} + n\pi\)) |
Continuous on domain |
| Power Functions \(x^n\) |
Depends on \(n\) | Continuous on domain |
💡 AP® Exam Shortcut: When asked to verify continuity of a polynomial, exponential, or trig function, you can simply state: "Since \(f(x)\) is a [polynomial/exponential/etc.], it is continuous on its entire domain. The interval [specify] is within this domain, therefore \(f\) is continuous on [interval]."
📖 Comprehensive Worked Examples
Example 1: Polynomial Function
Problem: Show that \(f(x) = 2x^3 - 5x + 1\) is continuous on \([-2, 3]\)
Solution:
- Identify function type: \(f(x)\) is a polynomial
- Apply continuity theorem: Polynomials are continuous everywhere
- Check interval: \([-2, 3]\) is within \((-\infty, \infty)\) ✓
- Conclusion: Since \(f(x)\) is a polynomial, it is continuous on \([-2, 3]\)
Answer: Continuous on \([-2, 3]\) ✓
Example 2: Rational Function
Problem: Determine if \(g(x) = \frac{x + 3}{x - 1}\) is continuous on \((1, \infty)\)
Solution:
- Check domain: Denominator = 0 when \(x = 1\)
- Domain of \(g(x)\): All real numbers except \(x = 1\)
- Check interval: \((1, \infty)\) excludes \(x = 1\) (open at that endpoint)
- For \(x > 1\): Denominator \(x - 1 > 0\), so \(g(x)\) is defined
- Apply theorem: Rational functions are continuous on their domain
- Conclusion: Since \((1, \infty) \subset \text{domain of } g\), \(g(x)\) is continuous on \((1, \infty)\)
Answer: Continuous on \((1, \infty)\) ✓
Example 3: Piecewise Function
Problem: Is \(h(x) = \begin{cases} x^2 & x \leq 1 \\ 2x - 1 & x > 1 \end{cases}\) continuous on \([0, 3]\)?
Solution:
- Check each piece:
- For \(x \leq 1\): \(x^2\) is a polynomial → continuous
- For \(x > 1\): \(2x - 1\) is a polynomial → continuous
- Check boundary at \(x = 1\):
- \(h(1) = 1^2 = 1\) (from left piece)
- \(\lim_{x \to 1^-} h(x) = \lim_{x \to 1^-} x^2 = 1\)
- \(\lim_{x \to 1^+} h(x) = \lim_{x \to 1^+} (2x-1) = 2(1) - 1 = 1\)
- Since \(\lim_{x \to 1^-} = \lim_{x \to 1^+} = h(1) = 1\) ✓
- Check endpoints of \([0, 3]\):
- At \(x = 0\): \(h(0) = 0\), \(\lim_{x \to 0^+} x^2 = 0\) ✓
- At \(x = 3\): \(h(3) = 2(3) - 1 = 5\), \(\lim_{x \to 3^-} (2x-1) = 5\) ✓
- Conclusion: All continuity conditions satisfied
Answer: Continuous on \([0, 3]\) ✓
Example 4: Function with Discontinuity
Problem: Is \(k(x) = \frac{x^2 - 4}{x - 2}\) continuous on \([0, 4]\)?
Solution:
- Check domain: Denominator = 0 when \(x = 2\)
- \(x = 2\) is in \([0, 4]\): This is a problem!
- At \(x = 2\): \(k(2)\) is undefined (\(\frac{0}{0}\))
- Since \(k(x)\) is not defined at \(x = 2\): Cannot be continuous there
- Conclusion: \(k(x)\) is NOT continuous on \([0, 4]\) because it has a removable discontinuity at \(x = 2\)
Answer: NOT continuous on \([0, 4]\) ✗
However: \(k(x)\) IS continuous on \([0, 2) \cup (2, 4]\)
Example 5: Logarithmic Function
Problem: On what intervals is \(f(x) = \ln(3x)\) continuous?
Solution:
- Find domain: Natural log requires \(3x > 0\)
- Solve: \(x > 0\)
- Domain: \((0, \infty)\)
- Apply theorem: Logarithmic functions are continuous on their domain
- Conclusion: \(f(x) = \ln(3x)\) is continuous on \((0, \infty)\)
Answer: Continuous on \((0, \infty)\)
🧩 Special Focus: Piecewise Functions
Key Strategy for Piecewise Functions:
- Check each piece separately — Most pieces are polynomials or other standard functions (continuous on their intervals)
- Focus on the boundary points — Where the function changes definition, you MUST verify:
- \(f(\text{boundary})\) is defined
- Left-hand limit = right-hand limit = function value
- Check interval endpoints (if closed) — Use appropriate one-sided limits
💡 Common Mistake: Students often forget to check the boundary points where piecewise functions change formulas. These are the most likely places for discontinuities to occur, so always verify continuity explicitly at these points!
🚫 Common Domain Restrictions to Check
Always check for these domain issues:
| Issue | Restriction | Example |
|---|---|---|
| Denominators | Cannot = 0 | \(\frac{1}{x-2}\) undefined at \(x=2\) |
| Even Roots | Radicand ≥ 0 | \(\sqrt{x-3}\) needs \(x \geq 3\) |
| Logarithms | Argument > 0 | \(\ln(x)\) needs \(x > 0\) |
| Tangent/Secant | Avoid odd multiples of \(\frac{\pi}{2}\) | \(\tan(x)\) undefined at \(\pm\frac{\pi}{2}, \pm\frac{3\pi}{2}\), etc. |
| Cotangent/Cosecant | Avoid multiples of \(\pi\) | \(\csc(x)\) undefined at \(0, \pm\pi, \pm2\pi\), etc. |
💡 Tips, Tricks & Strategies
✅ Essential Tips
- Know your interval notation: Open vs. closed determines whether you check endpoints
- Use function family properties: Cite theorems for polynomials, trig, exponential, etc.
- Focus on problem points: Check where denominators = 0, radicals negative, logs non-positive
- Piecewise → check boundaries: Always verify continuity where formula changes
- One-sided limits at endpoints: For closed intervals, use appropriate one-sided limits
- State your reasoning: On AP® exams, justify why a function is/isn't continuous
🎯 Quick Decision Tree
Is the function continuous on interval I?
- Is it a standard function? (polynomial, trig, exp, log)
- YES → Check if interval is within domain → If yes, continuous ✓
- NO → Continue to step 2
- Is it rational or has domain restrictions?
- Find problem points (denominator = 0, etc.)
- Are problem points in the interval? If yes, NOT continuous ✗
- If no problem points in interval → continuous ✓
- Is it piecewise?
- Check each piece (usually continuous)
- Check boundaries with one-sided limits
- If all match → continuous ✓
❌ Common Mistakes to Avoid
- Mistake 1: Forgetting to check if the interval is open or closed
- Mistake 2: Not verifying one-sided limits at endpoints of closed intervals
- Mistake 3: Assuming piecewise functions are automatically continuous at boundaries
- Mistake 4: Missing domain restrictions (denominators = 0, negative under square roots)
- Mistake 5: Not checking BOTH one-sided limits at piecewise boundaries
- Mistake 6: Citing "continuous everywhere" for rational functions (they have restrictions!)
- Mistake 7: Checking wrong endpoints for half-open intervals
- Mistake 8: Not justifying your answer with proper mathematical reasoning
📊 Quick Comparison: Point vs. Interval Continuity
| Aspect | Continuity at a Point | Continuity on an Interval |
|---|---|---|
| Definition | Three conditions at \(x = c\) | Continuous at every point in interval |
| What to check | One specific point | All points (or cite theorem) |
| Endpoints | N/A (single point) | Must check if interval is closed |
| Limits needed | Two-sided limit at \(c\) | Interior: two-sided; Endpoints: one-sided |
| AP® Exam | Verify 3 conditions explicitly | Identify intervals, cite theorems, check boundaries |
📝 Practice Problems
Determine if each function is continuous on the given interval:
- \(f(x) = x^3 - 2x + 5\) on \([-1, 4]\)
- \(g(x) = \frac{x+1}{x-3}\) on \((3, \infty)\)
- \(h(x) = \begin{cases} x^2 & x < 2 \\ 4x & x \geq 2 \end{cases}\) on \([0, 5]\)
- \(k(x) = \sqrt{x-1}\) on \([1, 10]\)
Answers:
- YES — Polynomial, continuous everywhere
- YES — Rational, continuous on domain; \(x = 3\) not in \((3, \infty)\)
- NO — At \(x = 2\): left limit = 4, right limit = 8 (jump discontinuity)
- YES — Square root continuous for \(x \geq 1\); check endpoint: \(\lim_{x \to 1^+} \sqrt{x-1} = 0 = f(1)\) ✓
✏️ AP® Exam Success Tips
What the AP® Exam Expects:
- Identify intervals of continuity: State clearly which intervals work
- Justify using theorems: Cite "polynomial continuous everywhere," etc.
- Check domain explicitly: For rational/log/root functions, find restrictions
- Verify boundaries for piecewise: Show one-sided limit calculations
- State endpoint conditions: For closed intervals, verify one-sided limits
- Use proper interval notation: Open vs. closed matters!
- Explain discontinuities: If not continuous, state why and where
Common FRQ Formats:
- "On what interval(s) is f continuous? Justify your answer."
- "Is f continuous on [a, b]? Use the definition of continuity to justify."
- "Find the values of k that make f continuous on the given interval." (piecewise)
- "Use the graph to identify all intervals where f is continuous."
⚡ Quick Reference Card
| Interval Type | Notation | What to Check |
|---|---|---|
| Open | \((a, b)\) | Interior points only (NOT endpoints) |
| Closed | \([a, b]\) | Interior + one-sided limits at BOTH endpoints |
| Half-Open Left | \([a, b)\) | Interior + right-limit at \(a\) only |
| Half-Open Right | \((a, b]\) | Interior + left-limit at \(b\) only |
| Infinite | \((a, \infty)\), etc. | All interior; check finite endpoint if closed |
🔗 Why This Unit Matters
Unit 1.12 is critical for:
- Unit 1.13: Removing discontinuities (need to identify continuous intervals first)
- Unit 1.14: Intermediate Value Theorem (requires continuity on closed interval)
- Unit 2: Differentiability (function must be continuous to be differentiable)
- Unit 4: Mean Value Theorem (requires continuity on closed interval)
- Unit 6: Fundamental Theorem of Calculus (requires continuity for definite integrals)
- Throughout calculus: Many theorems require continuity on intervals!
Remember: A function is continuous on an interval if it's continuous at every point in that interval. For closed intervals, don't forget to verify one-sided continuity at the endpoints! Use the "pencil test"—can you draw the entire graph over the interval without lifting your pencil? Master interval notation, cite theorems for standard functions, and always check boundaries for piecewise functions. This foundation prepares you for the Intermediate Value Theorem and beyond! 🎯✨