Continuity and Discontinuity of Functions

The video tutorial explains the concept of continuity for functions at a specific point, using x=3 as an example. It discusses the conditions under which a function is continuous, emphasizing the need for the function to be defined at that point and for the limit to equal the function's value. The tutorial evaluates two functions: G(x) = ln(x-3), which is not continuous at x=3 due to being undefined, and F(x) = e^(x-3), which is continuous for all real numbers. Visual aids are used to illustrate the behavior of these functions and their continuity.

What is the condition for a function to be continuous at a point?

• The function must be differentiable at that point.

• The function must be defined and the limit must exist and equal the function's value at that point.

• The function must be increasing at that point.

• The function must be decreasing at that point.

Why is the function G(x) = ln(x - 3) not continuous at x = 3?

• Because ln(x - 3) is always defined.

• Because ln(x - 3) is not defined at x = 3.

• Because ln(x - 3) is a linear function.

• Because ln(x - 3) is a polynomial function.

What happens to the function G(x) = ln(x - 3) at x = 3?

• It is defined and continuous.

• It is not defined and thus not continuous.

• It is continuous but not defined.

• It is defined but not continuous.

What is the value of f(3) for the function f(x) = e^(x - 3)?

• 0

• 1

• e

• 3

Why is the function f(x) = e^(x - 3) continuous at x = 3?

• Because it is not defined at x = 3.

• Because the limit as x approaches 3 equals f(3).

• Because it is a polynomial function.

• Because it is a logarithmic function.

Which of the following statements is true about f(x) = e^(x - 3)?

• It is not continuous for any real number.

• It is continuous for all real numbers.

• It is not defined for any real number.

• It is continuous only at x = 3.

How does the graph of G(x) = ln(x - 3) behave near x = 3?

• It has a discontinuity and is not defined at x = 3.

• It is a straight line.

• It is continuous and smooth.

• It is a parabola.

What is the visual difference between the graphs of G(x) and f(x)?

• G(x) is continuous, f(x) is not.

• Both are continuous at x = 3.

• f(x) is continuous, G(x) is not.

• Both are not continuous at x = 3.

What does the graph of f(x) = e^(x - 3) look like?

• It has jumps and gaps.

• It is a smooth curve without jumps or gaps.

• It is a straight line.

• It is a parabola.

What is the significance of the dotted line in the graph of G(x)?

• It indicates where the function is continuous.

• It represents the function's minimum value.

• It marks the point of discontinuity.

• It shows the function's maximum value.