KALKULUS | KONSEP KEKONTINUAN FUNGSI DI SATU TITIK

Introduction

This tutorial is designed to provide a clear understanding of the concept of continuity of functions at a point, a fundamental topic in calculus. Understanding this concept is crucial for students in various fields such as science, engineering, and business, as it lays the groundwork for more advanced mathematical thinking and applications.

Step 1: Understanding Continuity at a Point

Continuity at a point means that a function behaves predictably around that point. To determine if a function is continuous at a point (c), three conditions must be met:

1. Function Value Exists: The function (f(c)) must be defined.
2. Limit Exists: The limit of the function as it approaches (c) must exist:
   • ( \lim_{x \to c} f(x) ) must exist.
3. Limit Equals Function Value: The limit must equal the function's value at that point:
   • ( \lim_{x \to c} f(x) = f(c) )

Practical Advice

• To check these conditions, always start by evaluating the function at point (c).
• If the function is not defined at (c), it is automatically discontinuous.

Step 2: Exploring Left-Hand and Right-Hand Limits

To further analyze continuity, we look at the left-hand limit and right-hand limit of the function at point (c):

1. Left-Hand Limit:

   • This is the value the function approaches as (x) approaches (c) from the left:
   • ( \lim_{x \to c^-} f(x) )

2. Right-Hand Limit:

   • This is the value the function approaches as (x) approaches (c) from the right:
   • ( \lim_{x \to c^+} f(x) )

Practical Advice

• A function is continuous at (c) if both the left-hand limit and right-hand limit exist and are equal to each other and to (f(c)):
  • ( \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c) )

Step 3: Identifying Types of Discontinuity

If any of the conditions for continuity are not satisfied, the function is discontinuous. There are three types of discontinuities to be aware of:

1. Removable Discontinuity:

   • Occurs when a single point is undefined or different from the limit.

2. Jump Discontinuity:

   • Occurs when the left-hand and right-hand limits exist but are not equal.

3. Infinite Discontinuity:

   • Occurs when the function approaches infinity at a certain point.

Practical Advice

• Graphing the function can help visualize where discontinuities occur.
• Pay attention to piecewise functions, as they often exhibit jump discontinuities.

Step 4: Practical Examples

To solidify your understanding, work through examples of functions with known continuity properties:

1. Example Function: ( f(x) = \frac{x^2 - 1}{x - 1} )

   • Identify whether (f(1)) is defined and assess limits from both sides.

2. Example Function: ( f(x) = \begin{cases} 1 & \text{if } x < 2 \ 3 & \text{if } x \geq 2 \end{cases} )

   • Analyze left-hand and right-hand limits at (x = 2).

Practical Advice

• Always simplify functions before checking continuity.
• Review your findings with a graph for better insight.

Conclusion

Understanding the concept of continuity at a point is essential for mastering calculus. Remember to check the function's definition, analyze limits from both sides, and identify any discontinuities. Practicing with different functions will reinforce these concepts and prepare you for more complex calculus topics. For further learning, consider exploring limits and derivatives as they build upon the foundation of continuity.