KALKULUS  KONSEP KEKONTINUAN FUNGSI DI SATU TITIK
Table of Contents
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:
 Function Value Exists: The function (f(c)) must be defined.
 Limit Exists: The limit of the function as it approaches (c) must exist:
 ( \lim_{x \to c} f(x) ) must exist.
 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 LeftHand and RightHand Limits
To further analyze continuity, we look at the lefthand limit and righthand limit of the function at point (c):

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

RightHand 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 lefthand limit and righthand 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:

Removable Discontinuity:
 Occurs when a single point is undefined or different from the limit.

Jump Discontinuity:
 Occurs when the lefthand and righthand limits exist but are not equal.

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:

Example Function: ( f(x) = \frac{x^2  1}{x  1} )
 Identify whether (f(1)) is defined and assess limits from both sides.

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