Continuity of Functions
Understanding continuous functions and their properties
Interactive Continuity Visualization
Welcome to the Continuity Explorer
This interactive visualization helps you understand continuity and the three conditions that must be satisfied for a function to be continuous at a point. Explore six different function types to see various continuity scenarios.
What you can explore:
- Continuous functions - See how all three conditions are satisfied (f(x) = x²)
- Removable discontinuity - Limit exists but f(a) is undefined (hole)
- Jump discontinuity - Left and right limits differ, so limit doesn't exist
- Infinite discontinuity - Function approaches infinity, limit doesn't exist
- Oscillating discontinuity - Function oscillates infinitely, no limit exists
- Piecewise functions - Limit exists but doesn't equal function value
How to Use This Visualization
Interactive Features:
- • Select Function Type - Choose from 6 different functions to explore various continuity scenarios
- • Adjust Point a - Use the slider to test continuity at different points
- • Toggle Visualizations - Show/hide left/right approach, limit point, and continuity conditions
- • View Conditions - See real-time evaluation of all three continuity conditions
What You'll See:
- • Function Curve (colored) - The original function with proper discontinuity handling
- • Point a (red) - The point where continuity is being tested
- • Function Value f(a) - Shown in red (or "undefined" if not defined)
- • Limit Point (green) - The limit value when it exists
- • Left/Right Approach (blue/amber) - Points approaching from left and right
- • Continuity Conditions Panel - Real-time check of all three conditions with ✓ or ✗
All three conditions for continuity are satisfied
Adjust to test continuity at different points
Three Conditions for Continuity at x = a
Condition 1: f(a) is defined
f(1.00) = 1.00
Condition 2: limx→a f(x) exists
lim = 1.00
Condition 3: limx→a f(x) = f(a)
Limit equals function value
✓ Function is CONTINUOUS at x = 1.00
None - function is continuous
Function Information
Point a: 1.00
f(a): 1.00
Limit: 1.00
Continuous: Yes
Key Concepts
Continuity: A function is continuous at a point if you can draw it without lifting your pencil.
Discontinuity Types: Removable (hole), Jump (break), Infinite (vertical asymptote), Oscillating.
All Three Conditions: Must be satisfied for continuity. If any fails, the function is discontinuous.
Definition of Continuity
A function f is continuous at x = a if and only if:
- f(a) is defined
- limx→a f(x) exists
- limx→a f(x) = f(a)
Lessons
Individual learning units
Lessons coming soon