Piecewise Functions and Continuity

Professor Dave explains continuity in functions, highlighting continuous and discontinuous functions. He discusses asymptotes, limits, and discontinuities, using examples like 1/(x-1) and x^2-1/(x-1). The video also covers piecewise functions, demonstrating how they are evaluated differently based on input values.

What is a characteristic of a continuous function?

• It has gaps in its graph.

• It can be drawn without lifting the pencil.

• It is always a straight line.

• It has undefined points.

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

• It equals one.

• It becomes undefined.

• It equals infinity.

• It becomes zero.

What is an asymptote in the context of functions?

• A line that the function never touches.

• A point where the function is zero.

• A point where the function is maximum.

• A line that the function crosses.

What is a jump discontinuity?

• A point where the function equals zero.

• A continuous curve.

• A sudden change in the function's value.

• A point where the function is undefined.

In the function X^2 - 1 over X - 1, what happens at X=1?

• There is a hole in the graph.

• The function is continuous.

• The function equals zero.

• The function equals one.

How can a piecewise function be defined?

• By a single point.

• By a constant value for all X.

• By different equations for different intervals of X.

• By a single equation for all X.

What does an open circle in a graph of a piecewise function indicate?

• The point is included in the function.

• The function is undefined at that point.

• The point is not included in the function.

• The function is zero at that point.

What does a closed circle in a graph of a piecewise function indicate?

• The point is not included in the function.

• The point is included in the function.

• The function is zero at that point.

• The function is undefined at that point.

What is the value of the piecewise function when X is less than -2 in the given example?

• Undefined

• 0

• 5

• X - 1

What is the value of the piecewise function when X is greater than or equal to -2 in the given example?

• X - 1

• X + 1

• 5

• Undefined