Understanding Continuity and Discontinuity in Piecewise Functions

The video tutorial explains how to determine the points of discontinuity in a piecewise function using the limit definition of continuity. It covers the conditions for continuity at a point and applies them to a specific piecewise function. The tutorial tests for continuity at x=0 and x=2, showing that the function is continuous at x=0 but has a jump discontinuity at x=2. The results are verified using a graph of the function.

What are the three conditions for a function to be continuous at a point?

• The function is undefined, the limit does not exist, and the limit equals the function value.

• The function is defined, the limit exists, and the limit equals the function value.

• The function is defined, the limit does not exist, and the limit does not equal the function value.

• The function is undefined, the limit exists, and the limit equals the function value.

Which function rule is used for x less than or equal to 0 in the piecewise function?

• f(x) = 4x + 3

• f(x) = -3e^x + 6

• f(x) = ln(x + 1)

• f(x) = x^2 + 1

At which points does the piecewise function potentially have discontinuities?

• x = 0 and x = 2

• x = 1 and x = 3

• x = -1 and x = 1

• x = 2 and x = 3

What is the value of f(0) for the piecewise function?

• 0

• 6

• -3

• 3

What is the limit of f(x) as x approaches 0 from the left?

• 6

• -3

• 3

• 0

What is the limit of f(x) as x approaches 0 from the right?

• 3

• 0

• 6

• -3

What is the approximate value of f(2) for the piecewise function?

• 1.0986

• 3

• -16.1672

• 0

What is the limit of f(x) as x approaches 2 from the left?

• 0

• -16.1672

• 1.0986

• 3

What is the limit of f(x) as x approaches 2 from the right?

• 3

• 1.0986

• -16.1672

• 0

What type of discontinuity is present at x = 2?

• Oscillating discontinuity

• Infinite discontinuity

• Removable discontinuity

• Jump discontinuity