Three-Step Continuity Test

Introduction

• The lecture describes how to determine if a function is continuous at a certain point using the three-step continuity test.

The Test Steps

Step 1: Check if the function is defined at the point

• Function must be defined at the point a, meaning f(a) must exist and return a specific value.

Step 2: Check if the limit exists as x approaches a

• The limit of f(x) as x approaches a must exist.
• This involves verifying that the left-sided limit (as x approaches a from the left) is equal to the right-sided limit (as x approaches a from the right).

Step 3: Verify that the limit equals the function's value at the point

• The limit of f(x) as x approaches a must be equal to f(a).

Application of Test

• Example 1 (Continuous at x=2):

  • When x=2, the function is split into segments based on x value ranges.
  • Using the three steps, it's proven that the function is continuous at x=2 because all conditions of the test are met.

• Example 2 (Discontinuous at x=3):

  • Shows discontinuity at x=3 because the limits from the left and right do not match.
  • The type of discontinuity presented is a jump discontinuity, indicating a non-removable discontinuity.

• Example 3 (Discontinuous at x=-1):

  • Discontinuous due to the difference between the limit of f(x) as x approaches -1 and the value of f(-1).
  • Presents a removable discontinuity, known as a hole, because the limit exists but doesn’t match the function's value at the point.

Types of Discontinuities

• Jump Discontinuity: When the left and right limits do not match.
• Removable Discontinuity (Hole): When limits match but do not equal f(a).
• Infinite Discontinuity: Occurs if the limit values approach infinity, indicating an unbounded behavior near the point.