📐 Concept diagram

04-02 - Continuity

Phase: 4 | Subject: 04-02 Prerequisites: 04-01-limits.md Next subject: 04-03-the-derivative.md

Learning Objectives

By the end of this subject, you will be able to:

1. Define continuity at a point and on an interval
2. Identify types of discontinuities
3. Apply the Intermediate Value Theorem
4. Apply the Extreme Value Theorem

Core Content

Definition of Continuity

A function f(x) is continuous at x = a if: 1. f(a) is defined 2. $lim(x→a)$ f(x) exists 3. $lim(x→a)$ f(x) = f(a)

Intuitive: You can draw the graph at x = a without lifting your pen.

Continuous on an interval: Continuous at every point in the interval.

Types of Discontinuity

1. Removable Discontinuity

Limit exists, but f(a) is undefined or different from the limit. Fix: Redefine f(a) to equal the limit.

2. Jump Discontinuity

Left-hand and right-hand limits exist but are different. Example: Step function, floor function

3. Infinite Discontinuity

Function approaches ±∞ as x approaches a. Example: f(x) = 1/x at x = 0

Continuous Functions

Polynomials: Continuous everywhere Rational functions: Continuous wherever defined (denominator ≠ 0) Exponential functions: Continuous everywhere Logarithms: Continuous on their domain (x > 0) Trigonometric functions: Continuous on their domains Compositions: If f and g are continuous, so are f∘g, f+g, f-g, f·g, f/g (where g ≠ 0)

Intermediate Value Theorem (IVT)

⚠️ THIS IS CRITICAL — the IVT is used to prove that equations have solutions without actually solving them. It underpins root-finding algorithms (bisection method) and is essential for proving many calculus theorems.

If f is continuous on [a, b] and k is between f(a) and f(b), then there exists c ∈ [a, b] such that f(c) = k.

Applications: - Proving roots exist - Finding where a function equals a specific value

Example: Show x³ - x - 1 = 0 has a root between 1 and 2. f(1) = -1, f(2) = 5. By IVT, since -1 < 0 < 5, there's a root in (1, 2).

Extreme Value Theorem (EVT)

If f is continuous on [a, b], then f attains both a maximum and a minimum value on [a, b].

Key: Requires a CLOSED interval [a, b].

Example: f(x) = x² on [-1, 2] Continuous on closed interval. Minimum at x = 0: f(0) = 0. Maximum at x = 2: f(2) = 4.

Key Terms

• 04 02 Continuity
• Continuous Functions
• Correct: A)
• Correct: B)
• Correct: C)
• Definition of Continuity
• Example 1: Continuity check
• Example 2: IVT application
• Example 3: EVT and discontinuity
• Extreme Value Theorem (EVT)
• Infinite Discontinuity
• Intermediate Value Theorem (IVT)

Worked Examples

Example 1: Continuity check

f(x) = (x² - 4)/(x - 2) for x ≠ 2, f(2) = 5

At x = 2: - f(2) = 5 (defined) - $lim(x→2)$ (x² - 4)/(x - 2) = $lim(x→2)$ (x + 2) = 4 - Limit (4) ≠ f(2) (5)

Removable discontinuity at x = 2. Redefine f(2) = 4 to make continuous.

Example 2: IVT application

Show x = cos(x) has a solution.

f(x) = x - cos(x) f(0) = 0 - 1 = -1 f(π/2) = π/2 - 0 ≈ 1.57

By IVT, there's a root in (0, π/2).

Example 3: EVT and discontinuity

Does f(x) = 1/x attain a maximum and minimum on (0, 5]?

No. The interval is NOT closed on the left — (0, 5] excludes 0. As x → 0⁺, f(x) → +∞, so there is no maximum. The minimum occurs at x = 5: f(5) = 1/5 = 0.2. EVT requires [a, b] (closed on both ends).

Quiz

Q1: What does the concept of Continuous Functions primarily refer to in this subject?

A) A computational error related to Continuous Functions B) A visual representation of Continuous Functions C) The definition and application of Continuous Functions D) A historical anecdote about Continuous Functions

Correct: C)

• If you chose A: This is incorrect. Continuous Functions is defined as: the definition and application of continuous functions. The other options describe different aspects that are not the primary focus.
• If you chose B: This is incorrect. Continuous Functions is defined as: the definition and application of continuous functions. The other options describe different aspects that are not the primary focus.
• If you chose C: Continuous Functions is defined as: the definition and application of continuous functions. The other options describe different aspects that are not the primary focus. Correct!
• If you chose D: This is incorrect. Continuous Functions is defined as: the definition and application of continuous functions. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of Definition of Continuity?

A) It is used to definition of continuity in mathematical analysis B) It is used only in advanced research contexts C) It replaces all other methods in this domain D) It is primarily a historical notation system

Correct: A)

• If you chose A: Definition of Continuity serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose B: This is incorrect. Definition of Continuity serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose C: This is incorrect. Definition of Continuity serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose D: This is incorrect. Definition of Continuity serves the purpose described in the correct answer. The other options misrepresent its role.

Q3: Which statement about Extreme Value Theorem (EVT) is TRUE?

A) Extreme Value Theorem (EVT) is not related to this subject B) Extreme Value Theorem (EVT) is a fundamental concept covered in this subject C) Extreme Value Theorem (EVT) is an advanced topic beyond this subject's scope D) Extreme Value Theorem (EVT) is mentioned only as a historical footnote

Correct: B)

• If you chose A: This is incorrect. Extreme Value Theorem (EVT) is a fundamental concept covered in this subject. This subject covers Extreme Value Theorem (EVT) as part of its core content.
• If you chose B: Extreme Value Theorem (EVT) is a fundamental concept covered in this subject. This subject covers Extreme Value Theorem (EVT) as part of its core content. Correct!
• If you chose C: This is incorrect. Extreme Value Theorem (EVT) is a fundamental concept covered in this subject. This subject covers Extreme Value Theorem (EVT) as part of its core content.
• If you chose D: This is incorrect. Extreme Value Theorem (EVT) is a fundamental concept covered in this subject. This subject covers Extreme Value Theorem (EVT) as part of its core content.

Q4: Based on the worked examples in this subject, what is the correct result?

A) ** No. f(0) is undefined. Infinite discontinuity. B) The inverse of the correct answer C) A different result from a common mistake D) An unrelated numerical value

Correct: A)

• If you chose A: The worked examples show that the result is ** No. f(0) is undefined. Infinite discontinuity.. The other options represent common errors. Correct!
• If you chose B: This is incorrect. The worked examples show that the result is ** No. f(0) is undefined. Infinite discontinuity.. The other options represent common errors.
• If you chose C: This is incorrect. The worked examples show that the result is ** No. f(0) is undefined. Infinite discontinuity.. The other options represent common errors.
• If you chose D: This is incorrect. The worked examples show that the result is ** No. f(0) is undefined. Infinite discontinuity.. The other options represent common errors.

Q5: How are Extreme Value Theorem (EVT) and Infinite Discontinuity related?

A) Extreme Value Theorem (EVT) is the inverse of Infinite Discontinuity B) Extreme Value Theorem (EVT) and Infinite Discontinuity are closely related concepts C) Extreme Value Theorem (EVT) and Infinite Discontinuity are completely unrelated topics D) Extreme Value Theorem (EVT) is a special case of Infinite Discontinuity

Correct: B)

• If you chose A: This is incorrect. Both Extreme Value Theorem (EVT) and Infinite Discontinuity are covered in this subject as interconnected topics.
• If you chose B: Both Extreme Value Theorem (EVT) and Infinite Discontinuity are covered in this subject as interconnected topics. Correct!
• If you chose C: This is incorrect. Both Extreme Value Theorem (EVT) and Infinite Discontinuity are covered in this subject as interconnected topics.
• If you chose D: This is incorrect. Both Extreme Value Theorem (EVT) and Infinite Discontinuity are covered in this subject as interconnected topics.

Q6: What is a common pitfall when working with Intermediate Value Theorem (IVT)?

A) The main error with Intermediate Value Theorem (IVT) is using it when it is not needed B) Intermediate Value Theorem (IVT) has no common misconceptions C) Intermediate Value Theorem (IVT) is always computed the same way in all contexts D) A common mistake is confusing Intermediate Value Theorem (IVT) with a similar concept

Correct: D)

• If you chose A: This is incorrect. Students often confuse Intermediate Value Theorem (IVT) with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose B: This is incorrect. Students often confuse Intermediate Value Theorem (IVT) with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose C: This is incorrect. Students often confuse Intermediate Value Theorem (IVT) with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose D: Students often confuse Intermediate Value Theorem (IVT) with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!

Q7: When should you apply Types Of Discontinuity?

A) Types Of Discontinuity is not practically useful B) Avoid Types Of Discontinuity unless explicitly instructed C) Use Types Of Discontinuity only in pure mathematics contexts D) Apply Types Of Discontinuity to solve problems in this subject's domain

Correct: D)

• If you chose A: This is incorrect. Types Of Discontinuity is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: This is incorrect. Types Of Discontinuity is a practical tool used throughout this subject to solve relevant problems.
• If you chose C: This is incorrect. Types Of Discontinuity is a practical tool used throughout this subject to solve relevant problems.
• If you chose D: Types Of Discontinuity is a practical tool used throughout this subject to solve relevant problems. Correct!

Practice Problems

1. Is f(x) = 1/x continuous at x = 0? Answer: No. f(0) is undefined. Infinite discontinuity.

2. Is f(x) = x² continuous at x = 3? Answer: Yes. Polynomials are continuous everywhere. f(3) = 9, limit = 9.

3. Type of discontinuity for f(x) = (x² - 1)/(x - 1) at x = 1? Answer: Removable. Factor: (x-1)(x+1)/(x-1) = x+1 (for x ≠ 1). Limit = 2, but f(1) undefined.

4. IVT: Does f(x) = x³ - 2x + 1 have a root in [0, 2]? Answer: f(0) = 1, f(2) = 5. Both positive. Can't conclude a root. Try [-2, 0]: f(-2) = -3, f(0) = 1. Root in [-2, 0].

5. EVT: Does f(x) = 1/x have a max/min on (0, 1]? Answer: No. Interval is not closed on the left. f(x) → ∞ as x → 0⁺.

Summary

Key takeaways:

• Continuous at a: f(a) defined, limit exists, limit = f(a)
• Continuous everywhere: polynomials, exponentials, sin, cos
• Discontinuities: removable, jump, infinite
• IVT: continuous on [a,b] hits every value between f(a) and f(b)
• EVT: continuous on [a,b] attains max and min

Pitfalls

• Forgetting the three conditions for continuity: Continuity at x = a requires ALL three: f(a) is defined, $lim(x→a)$ f(x) exists, and $lim(x→a)$ f(x) = f(a). Missing any one means discontinuous. Students often check only one or two conditions, especially overlooking the requirement that the limit must equal the function value.
• Confusing continuity with differentiability: Continuity does NOT imply differentiability. A function like f(x) = |x| is continuous everywhere but not differentiable at x = 0. The converse IS true: if a function is differentiable at a point, it is automatically continuous there.
• Misclassifying removable discontinuities: A removable discontinuity occurs when the LIMIT exists but f(a) is either undefined or not equal to the limit. The graph has a hole at x = a. If both one-sided limits exist but differ, it's a jump; if the function blows up to ±∞, it's infinite. Mixing these up leads to incorrect classification.
• Applying IVT without verifying the sign condition: The IVT guarantees a root in [a, b] only when f(a) and f(b) have OPPOSITE signs. If both are positive or both negative, the IVT says nothing about roots in that interval — there could still be a root, but the IVT doesn't guarantee it.
• Applying EVT on open or half-open intervals: The extreme value theorem requires continuity on a CLOSED interval [a, b]. On (a, b] or (a, b), the function may fail to attain a max or min. For example, f(x) = 1/x on (0, 1] has no maximum because it grows without bound as x → 0⁺.

Next Steps

Next up: 04-03-the-derivative.md