Discontinuity and Asymptotes in Functions

Professor Dave explains continuity in functions, highlighting continuous functions that can be drawn without lifting a pencil. He introduces discontinuous functions, discussing undefined points and asymptotes, where functions approach but never touch certain lines. The video covers holes and jumps in functions, using examples to illustrate these concepts. Finally, piecewise functions are explored, showing how different function pieces are evaluated based on input values.

What is a characteristic of a continuous function?

• It is always a straight line.

• It can be drawn without lifting the pencil.

• It has undefined points.

• It has gaps.

Which of the following is true about discontinuous functions?

• They have no undefined points.

• They can be drawn without lifting the pencil.

• They are always parabolas.

• They may have gaps or undefined points.

What happens to the function 1/(x-1) at x=1?

• It becomes zero.

• It is undefined and has an asymptote.

• It equals one.

• It is defined and continuous.

What is an asymptote?

• A point where the function is undefined.

• A gap in the function.

• A line that the function approaches but never touches.

• A point where the function is zero.

How does the function 1/(x-1) behave as x approaches 1 from the left?

• It remains constant.

• It approaches zero.

• It approaches positive infinity.

• It approaches negative infinity.

What happens to the function 1/(x-1) as x approaches 1 from the right?

• It remains constant.

• It approaches positive infinity.

• It approaches zero.

• It approaches negative infinity.

What is a characteristic of a function with a hole?

• It is continuous everywhere.

• It is always a straight line.

• It has a missing point.

• It has an asymptote.

In the function (x^2-1)/(x-1), what happens at x=1?

• The function is continuous.

• The function equals zero.

• The function is undefined everywhere.

• There is a hole at x=1.

What is the domain of the function (x^2-1)/(x-1)?

• All real numbers except x=1.

• All real numbers.

• Only negative numbers.

• Only positive numbers.

What is a piecewise function?

• A function that is always linear.

• A function that is continuous everywhere.

• A function defined by multiple expressions over different intervals.

• A function with no defined domain.