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Unit 1.10 – Exploring Types of Discontinuities

AP® Calculus AB & BC | Formula Reference Sheet

Core Concept: A discontinuity is any point where a function "breaks"—where you'd have to lift your pencil while drawing the graph. Understanding the three types of discontinuities (removable, jump, and infinite) is essential for analyzing function behavior and mastering continuity concepts in AP® Calculus!

🎯 What is Continuity? (Quick Review)

The Three Conditions for Continuity

A function \(f(x)\) is continuous at \(x = a\) if and only if ALL three conditions hold:

  1. \(f(a)\) is defined — The function has a value at \(x = a\)
  2. \(\lim_{x \to a} f(x)\) exists — Both one-sided limits exist and are equal
  3. \(\lim_{x \to a} f(x) = f(a)\) — The limit equals the function value

If ANY of these conditions fail, the function is discontinuous at \(x = a\).

📝 The Pencil Test: If you can draw the entire graph without lifting your pencil, the function is continuous. If you must lift your pencil at any point, that's where a discontinuity occurs!

📚 The Three Types of Discontinuities

THE THREE TYPES OF DISCONTINUITIES

  1. Removable Discontinuity (Hole)
  2. Jump Discontinuity (Step/Gap)
  3. Infinite Discontinuity (Vertical Asymptote)

🔵 Type 1: Removable Discontinuity (Hole)

REMOVABLE DISCONTINUITY

Definition:

A function has a removable discontinuity at \(x = a\) when the limit exists, but either:

  • \(f(a)\) is undefined, OR
  • \(f(a)\) is defined but \(f(a) \neq \lim_{x \to a} f(x)\)

Mathematical Condition:

\[ \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L \] \[ \text{but } f(a) \neq L \text{ or } f(a) \text{ is undefined} \]

Graphical Appearance:

  • An open circle (hole) at \((a, L)\)
  • The curve approaches \((a, L)\) from both sides
  • Sometimes a filled dot exists at a different y-value

Algebraic Cause:

  • Common factor in numerator and denominator that cancels
  • Piecewise function with a "blip" at one point

Why "Removable"?

You can "fix" the discontinuity by redefining \(f(a) = L\). This makes the function continuous!

Example 1: Classic Removable Discontinuity

Function: \(f(x) = \frac{x^2 - 4}{x - 2}\)

Solution:

  1. Check \(f(2)\): \(\frac{0}{0}\) — undefined! ✗
  2. Factor: \(\frac{(x-2)(x+2)}{x-2}\)
  3. Cancel: \(f(x) = x + 2\) for \(x \neq 2\)
  4. Find limit: \(\lim_{x \to 2} f(x) = \lim_{x \to 2} (x+2) = 4\)
  5. Conclusion: Removable discontinuity (hole) at \((2, 4)\)

Graph: The line \(y = x + 2\) with an open circle at \((2, 4)\)

To remove: Define \(g(x) = \begin{cases} \frac{x^2-4}{x-2} & x \neq 2 \\ 4 & x = 2 \end{cases}\) — now continuous!

Example 2: Piecewise with Hole

Function: \(f(x) = \begin{cases} x^2 & x \neq 3 \\ 5 & x = 3 \end{cases}\)

Analysis:

  • \(\lim_{x \to 3} x^2 = 9\)
  • \(f(3) = 5\)
  • Since \(9 \neq 5\), removable discontinuity at \(x = 3\)

Graph: Parabola with open circle at \((3, 9)\) and filled dot at \((3, 5)\)

🟠 Type 2: Jump Discontinuity (Step/Gap)

JUMP DISCONTINUITY

Definition:

A function has a jump discontinuity at \(x = a\) when both one-sided limits exist and are finite, but they are NOT equal.

Mathematical Condition:

\[ \lim_{x \to a^-} f(x) = L_1 \quad \text{and} \quad \lim_{x \to a^+} f(x) = L_2 \] \[ \text{where } L_1 \neq L_2 \text{ (both finite)} \]

Graphical Appearance:

  • The graph "jumps" from one y-value to another
  • Vertical gap between the two branches
  • One side has an open circle, the other might have a filled dot

Common Causes:

  • Piecewise functions with different formulas on each side
  • Step functions (like greatest integer function)
  • Functions with different behavior at boundaries

Why NOT Removable?

The two-sided limit does not exist (left ≠ right), so you cannot fix it by redefining a single point!

Example 3: Classic Jump Discontinuity

Function: \(h(x) = \begin{cases} x - 1 & x < 1 \\ 2 - x & x \geq 1 \end{cases}\)

Solution:

  1. Left-hand limit: \(\lim_{x \to 1^-} h(x) = \lim_{x \to 1^-} (x-1) = 0\)
  2. Right-hand limit: \(\lim_{x \to 1^+} h(x) = \lim_{x \to 1^+} (2-x) = 1\)
  3. Compare: \(0 \neq 1\) — one-sided limits differ!
  4. Function value: \(h(1) = 2 - 1 = 1\) (from right piece)
  5. Conclusion: Jump discontinuity at \(x = 1\)

Jump size: \(|1 - 0| = 1\) unit

Graph: Left branch approaches \((1, 0)\) with open circle; right branch starts at \((1, 1)\) with filled dot

Example 4: Step Function

Function: \(f(x) = \begin{cases} 2 & x < 0 \\ -3 & x \geq 0 \end{cases}\)

Analysis:

  • \(\lim_{x \to 0^-} f(x) = 2\)
  • \(\lim_{x \to 0^+} f(x) = -3\)
  • \(2 \neq -3\) → Jump discontinuity
  • Jump size: \(|2 - (-3)| = 5\) units

🔴 Type 3: Infinite Discontinuity (Vertical Asymptote)

INFINITE DISCONTINUITY

Definition:

A function has an infinite discontinuity at \(x = a\) when at least one of the one-sided limits is infinite (\(\pm\infty\)).

Mathematical Condition:

\[ \lim_{x \to a^-} f(x) = \pm\infty \quad \text{or} \quad \lim_{x \to a^+} f(x) = \pm\infty \] \[ \text{(or both)} \]

Graphical Appearance:

  • Vertical asymptote at \(x = a\) (dashed vertical line)
  • Function "shoots up" to \(+\infty\) or "shoots down" to \(-\infty\)
  • Graph approaches the vertical line but never touches it

Common Causes:

  • Rational functions where denominator = 0 but numerator ≠ 0
  • After factoring and canceling, remaining zeros in denominator
  • Logarithmic functions at domain boundaries

Why NOT Removable?

The limit is infinite, not finite. You cannot "fix" infinity by redefining a point!

Example 5: Vertical Asymptote (Both Sides Same Direction)

Function: \(p(x) = \frac{3}{(x-2)^2}\)

Solution:

  1. Check denominator: \((x-2)^2 = 0\) when \(x = 2\)
  2. Left-hand limit: \(\lim_{x \to 2^-} \frac{3}{(x-2)^2} = +\infty\)
  3. Right-hand limit: \(\lim_{x \to 2^+} \frac{3}{(x-2)^2} = +\infty\)
  4. Both sides → \(+\infty\)
  5. Conclusion: Infinite discontinuity (vertical asymptote) at \(x = 2\)

Graph: U-shaped curve opening upward, approaching vertical line \(x = 2\)

Example 6: Vertical Asymptote (Opposite Directions)

Function: \(q(x) = \frac{1}{x-3}\)

Solution:

  1. Denominator zero: \(x = 3\)
  2. Left-hand limit: \(\lim_{x \to 3^-} \frac{1}{x-3} = -\infty\) (negative small number)
  3. Right-hand limit: \(\lim_{x \to 3^+} \frac{1}{x-3} = +\infty\) (positive small number)
  4. Conclusion: Infinite discontinuity at \(x = 3\)

Graph: Hyperbola with branches going to \(-\infty\) from left, \(+\infty\) from right

📊 Comparison Table: The Three Types

Feature Removable (Hole) Jump (Step) Infinite (Asymptote)
Two-sided limit Exists (finite) Does NOT exist Does NOT exist
Left-hand limit Exists = \(L\) Exists = \(L_1\) \(\pm\infty\) or finite
Right-hand limit Exists = \(L\) Exists = \(L_2\) (\(L_1 \neq L_2\)) \(\pm\infty\) or finite
Function value Undefined or ≠ limit May exist Undefined
Graph appearance Open circle (hole) Vertical gap/jump Vertical asymptote
Can it be fixed? YES (redefine point) NO NO
Common cause Canceled factor Piecewise mismatch Denominator = 0
Also called Point discontinuity Step discontinuity Essential discontinuity

🔍 How to Identify Each Type

Quick Identification Flowchart:

  1. Step 1: Compute \(\lim_{x \to a^-} f(x)\) and \(\lim_{x \to a^+} f(x)\)
  2. Step 2: Check if either limit is infinite
    • YES → Infinite discontinuity (vertical asymptote)
    • NO → Continue to Step 3
  3. Step 3: Are the two one-sided limits equal?
    • NO → Jump discontinuity
    • YES → Continue to Step 4
  4. Step 4: Does the limit equal \(f(a)\)?
    • NO (or \(f(a)\) undefined) → Removable discontinuity
    • YES → Function is continuous (no discontinuity!)

🧮 Algebraic Techniques for Each Type

For Removable Discontinuities

Strategy: Factor and Cancel

  1. Factor numerator and denominator completely
  2. Cancel common factors (these create the hole)
  3. Evaluate the simplified expression at \(x = a\)
  4. This gives you the limit value \(L\)
  5. The hole is at \((a, L)\)

For Jump Discontinuities

Strategy: Evaluate Piecewise Pieces

  1. Identify which formula applies for \(x < a\)
  2. Identify which formula applies for \(x > a\)
  3. Evaluate each piece as \(x \to a\)
  4. Compare the two values
  5. If different → jump discontinuity

For Infinite Discontinuities

Strategy: Check Denominator Zeros

  1. Find where denominator = 0
  2. Check if numerator ≠ 0 at that point
  3. Test one-sided limits (approach from left and right)
  4. Determine if each side goes to \(+\infty\) or \(-\infty\)
  5. Vertical asymptote at that x-value

📖 Comprehensive Worked Examples

Example 7: Identify the Type

Function: \(f(x) = \frac{x^2 - 9}{x^2 - 5x + 6}\)

Find all discontinuities and classify each type.

Solution:

  1. Factor numerator: \(x^2 - 9 = (x-3)(x+3)\)
  2. Factor denominator: \(x^2 - 5x + 6 = (x-2)(x-3)\)
  3. Rewrite: \(f(x) = \frac{(x-3)(x+3)}{(x-2)(x-3)}\)
  4. Cancel \((x-3)\): \(f(x) = \frac{x+3}{x-2}\) for \(x \neq 3\)
  5. Discontinuity at \(x = 3\):
    • Factor canceled → Removable discontinuity
    • \(\lim_{x \to 3} f(x) = \frac{6}{1} = 6\)
    • Hole at \((3, 6)\)
  6. Discontinuity at \(x = 2\):
    • Denominator = 0, numerator ≠ 0
    • \(\lim_{x \to 2^-} \frac{x+3}{x-2} = -\infty\)
    • \(\lim_{x \to 2^+} \frac{x+3}{x-2} = +\infty\)
    • Infinite discontinuity (vertical asymptote)

Answer:

  • Removable discontinuity at \(x = 3\)
  • Infinite discontinuity at \(x = 2\)

💡 Tips, Tricks & Memory Aids

✅ Essential Tips

  • Always check one-sided limits: They tell you everything about the discontinuity
  • Factor first: For rational functions, factor before identifying type
  • Canceled factors = holes: If a factor cancels, it creates a removable discontinuity
  • Remaining denominator zeros = asymptotes: After canceling, leftover zeros are vertical asymptotes
  • Piecewise → check boundaries: Always evaluate both pieces at boundary points
  • Use the three-condition test: Systematically check all continuity conditions

🎯 Memory Tricks

"RJI" Mnemonic:

  • Removable → Repairable (can fix by redefining)
  • Jump → Jump over a gap
  • Infinite → Infinity (shoots to \(\infty\))

"Hole, Step, Wall":

  • Hole = Removable (single missing point)
  • Step = Jump (step up or down)
  • Wall = Infinite (vertical wall/asymptote)

🔥 Quick Recognition Patterns

If you see...

  • \(\frac{0}{0}\) after substitution → Likely removable (try factoring)
  • \(\frac{\text{nonzero}}{0}\) after substitution → Infinite discontinuity
  • Piecewise with different formulas → Check for jump at boundaries
  • Common factor that cancels → Removable at that x-value
  • Floor/ceiling function \(\lfloor x \rfloor\) → Jump at every integer
  • Absolute value in piecewise → Possible jump where expression = 0

❌ Common Mistakes to Avoid

  • Mistake 1: Calling every discontinuity "removable"—not all can be fixed!
  • Mistake 2: Forgetting to check BOTH one-sided limits
  • Mistake 3: Confusing \(f(a)\) with \(\lim_{x \to a} f(x)\)—they can be different!
  • Mistake 4: Not factoring before identifying type in rational functions
  • Mistake 5: Assuming piecewise functions are always jump discontinuities
  • Mistake 6: Saying limit "equals infinity"—should say "approaches" or "is" infinity
  • Mistake 7: Not checking which circle is open/closed in piecewise graphs
  • Mistake 8: Thinking a function can have two types of discontinuities at the same point

✏️ AP® Exam Strategies

What the AP® Exam Expects:

  • Classify correctly: Know the three types and how to identify them
  • Justify with limits: Show one-sided limits to support your classification
  • Use proper terminology: Say "removable," "jump," or "infinite"—not just "discontinuous"
  • Check all three conditions: Reference the continuity definition explicitly
  • Draw accurate graphs: Mark holes with open circles, show asymptotes with dashed lines
  • State locations clearly: "At \(x = a\), the function has a [type] discontinuity"

Common FRQ Formats:

  1. "Identify and classify all discontinuities"
  2. "Is the function continuous at \(x = a\)? Justify your answer."
  3. "At which x-values does the function have a [type] discontinuity?"
  4. "Can the discontinuity be removed? If so, how?"
  5. "Sketch the graph showing all discontinuities"

Scoring Rubric Points:

  • Correctly computing one-sided limits
  • Proper classification of discontinuity type
  • Clear justification using continuity definition
  • Accurate graph with correct notation (open/closed circles)
  • Statement about removability (if applicable)

📝 Practice Problems

Classify each discontinuity:

  1. \(f(x) = \frac{x^2-16}{x-4}\) at \(x = 4\)
  2. \(g(x) = \begin{cases} 3x & x < 2 \\ x + 4 & x \geq 2 \end{cases}\) at \(x = 2\)
  3. \(h(x) = \frac{2x+1}{x-5}\) at \(x = 5\)
  4. \(p(x) = \frac{x^2-1}{x^2+x-2}\) at \(x = 1\)

Answers:

  1. Removable — factors to \(x + 4\), hole at \((4, 8)\)
  2. Jump — left limit = 6, right limit = 6... wait, they're equal! Continuous!
  3. Infinite — vertical asymptote at \(x = 5\)
  4. Removable — \((x-1)\) cancels, hole at \((1, 2/3)\)

⚡ Quick Reference Card

Type Limits Graph Fix?
Removable \(\lim_{x \to a^-} = \lim_{x \to a^+} = L\)
\(f(a) \neq L\)		 Hole at \((a,L)\) ✓ YES
Jump \(\lim_{x \to a^-} \neq \lim_{x \to a^+}\)
Both finite		 Vertical gap ✗ NO
Infinite At least one = \(\pm\infty\) Vertical asymptote ✗ NO

🔗 Why This Unit Matters

Unit 1.10 connects to:

  • Unit 1.11-1.16: More continuity topics (IVT, removing discontinuities)
  • Unit 2: Derivatives don't exist at discontinuities
  • Unit 3: Analyzing where functions are/aren't continuous
  • Unit 6: Integrals require continuity (or piecewise handling)
  • Throughout calculus: Understanding where functions break is essential!

Remember: The three types of discontinuities tell the story of how functions can "break." Removable discontinuities (holes) can be fixed by filling in one point. Jump discontinuities (steps) have a vertical gap that can't be bridged. Infinite discontinuities (vertical asymptotes) shoot to infinity and represent fundamental breaks. Master identifying these three types using one-sided limits, and you'll have a complete understanding of where and why functions are discontinuous! 🎯📊✨