Continuità delle Funzioni
Comprendere funzioni continue e le loro proprietà
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)
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