Continuity
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Definition of Continuity
Continuity at a Point:
A function f(x) is continuous at x = a if all three conditions are satisfied:
Three Conditions:
1. \( f(a) \) is defined (function exists at x = a)
2. \( \lim_{x \to a} f(x) \) exists (limit exists as x approaches a)
3. \( \lim_{x \to a} f(x) = f(a) \) (limit equals function value)
\[ \lim_{x \to a} f(x) = f(a) \]
Intuitive Understanding:
A function is continuous if you can draw its graph without lifting your pencil
• No holes in the graph
• No jumps or breaks
• No vertical asymptotes
2. One-Sided Continuity
Continuous from the Right:
f(x) is continuous from the right at x = a if:
\[ \lim_{x \to a^+} f(x) = f(a) \]
Continuous from the Left:
f(x) is continuous from the left at x = a if:
\[ \lim_{x \to a^-} f(x) = f(a) \]
Key Point:
A function is continuous at x = a if and only if it is continuous from both the left and the right at x = a
Two-sided continuity requires both one-sided limits to exist, be equal, and equal f(a)
3. Types of Discontinuity
1. Removable Discontinuity (Point/Hole):
A hole in the graph where the limit exists but doesn't equal the function value
Characteristics:
• \( \lim_{x \to a} f(x) \) exists
• Either f(a) is undefined OR \( \lim_{x \to a} f(x) \neq f(a) \)
• Can be "fixed" by redefining f(a)
Example:
\( f(x) = \frac{x^2-4}{x-2} \) at x = 2
Has a hole at (2, 4) because limit = 4 but f(2) undefined
2. Jump Discontinuity:
The function "jumps" from one value to another
Characteristics:
• Both one-sided limits exist and are finite
• \( \lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x) \)
• Two-sided limit does NOT exist
Example:
Step function or piecewise function with different values on each side
3. Infinite Discontinuity:
Function approaches infinity (vertical asymptote)
Characteristics:
• At least one one-sided limit is infinite
• \( \lim_{x \to a^-} f(x) = \pm\infty \) or \( \lim_{x \to a^+} f(x) = \pm\infty \)
• Vertical asymptote at x = a
Example:
\( f(x) = \frac{1}{x-3} \) at x = 3
Vertical asymptote at x = 3
4. Identifying Continuous Functions from Graphs
Visual Checklist:
✓ Function IS Continuous if:
• Graph is unbroken (no gaps)
• No holes (open circles)
• No jumps or breaks
• No vertical asymptotes
✗ Function is NOT Continuous if:
• Open circle (hole) in graph
• Sudden jump between two parts
• Vertical asymptote present
Examples of Continuous Functions:
• Polynomials: \( f(x) = x^2 + 3x - 5 \)
• Exponential functions: \( f(x) = e^x \)
• Sine and cosine: \( f(x) = \sin(x) \)
• Continuous on their entire domain
5. Continuity on an Interval
Open Interval (a, b):
f(x) is continuous on the open interval (a, b) if it is continuous at every point in the interval
Does not include endpoints a and b
Closed Interval [a, b]:
f(x) is continuous on the closed interval [a, b] if:
• f(x) is continuous at every point in (a, b)
• f(x) is continuous from the right at x = a: \( \lim_{x \to a^+} f(x) = f(a) \)
• f(x) is continuous from the left at x = b: \( \lim_{x \to b^-} f(x) = f(b) \)
6. Analyzing Points of Discontinuity
Steps to Analyze:
1. Identify the point: Find where discontinuity occurs
2. Check f(a): Is the function defined at x = a?
3. Find one-sided limits: Calculate \( \lim_{x \to a^-} f(x) \) and \( \lim_{x \to a^+} f(x) \)
4. Compare values: Are the limits equal? Do they equal f(a)?
5. Classify: Determine type of discontinuity
Decision Tree:
If \( \lim_{x \to a} f(x) \) exists:
→ And equals f(a): Continuous
→ But doesn't equal f(a): Removable Discontinuity
If \( \lim_{x \to a} f(x) \) does NOT exist:
→ One-sided limits are finite but unequal: Jump Discontinuity
→ At least one limit is infinite: Infinite Discontinuity
7. Properties of Continuous Functions
Operations Preserve Continuity:
If f and g are continuous at x = a, then:
• \( f + g \) is continuous at x = a
• \( f - g \) is continuous at x = a
• \( cf \) is continuous at x = a (c is constant)
• \( f \cdot g \) is continuous at x = a
• \( \frac{f}{g} \) is continuous at x = a (if g(a) ≠ 0)
8. Intermediate Value Theorem (IVT)
Statement:
If f is continuous on [a, b] and N is any number between f(a) and f(b), then there exists at least one number c in (a, b) such that f(c) = N
Practical Meaning:
A continuous function takes on every value between any two of its values
Application:
Used to prove that equations have solutions in certain intervals
9. Quick Reference Summary
Continuity Checklist:
| Type | Limit Exists? | Limit = f(a)? |
|---|---|---|
| Continuous | Yes | Yes |
| Removable | Yes | No |
| Jump | No (limits unequal) | N/A |
| Infinite | No (limit is ±∞) | N/A |
📚 Study Tips
✓ A continuous function has no breaks, holes, or jumps
✓ Check all three conditions: f(a) defined, limit exists, limit = f(a)
✓ Removable discontinuities can be "fixed" by redefining the function
✓ Polynomials and exponentials are continuous everywhere
✓ Always check one-sided limits at potential discontinuities