Understanding Continuous Functions

The video tutorial explains the concept of continuity in functions, emphasizing that a function is continuous over an interval if it has no jumps, gaps, or discontinuities. It describes different types of discontinuities, such as asymptotic and jump discontinuities, and provides a mathematical definition using limits. The tutorial also discusses examples of continuous functions, like e^x, which are defined for all real numbers without any gaps or jumps.

What is a key characteristic of a continuous function?

• It has gaps.

• It is only defined for positive numbers.

• It has jumps.

• It is connected without interruptions.

Which of the following is an example of a discontinuity?

• A smooth curve.

• A jump in the graph.

• A straight line.

• A constant function.

What type of discontinuity involves a gap where the function is not defined?

• Gap discontinuity

• Jump discontinuity

• Removable discontinuity

• Infinite discontinuity

What is a removable discontinuity?

• A point where the function is zero.

• A point where the function is infinite.

• A jump in the graph.

• A point where the function is not defined but can be redefined to make it continuous.

What must be true for a function to be continuous at a point?

• The limit must not exist at that point.

• The function must be undefined at that point.

• The function must have a jump at that point.

• The limit as x approaches the point must equal the function's value at that point.

Why is G(x) not continuous for all real numbers?

• It is only defined for integer values of x.

• It is not defined for negative values of x.

• It is not defined for positive values of x.

• It has a jump at x = 0.

What does it mean for a function to be defined for all real numbers?

• It is only defined for positive numbers.

• It is defined for both positive and negative numbers.

• It is only defined for negative numbers.

• It is only defined for integers.

Which function is continuous for all real numbers?

• 1/x

• sin(x)

• e^x

• x^2

Which of the following functions is not continuous for all real numbers?

• x^3

• 1/x

• cos(x)

• e^x

What is the graphical representation of e^x?

• A zigzag line.

• A line with a jump.

• A curve with gaps.

• A smooth, continuous curve.