Continuous, Discontinuous, and Piecewise Functions

The video tutorial covers continuous and discontinuous functions, explaining how continuous functions have no gaps and can be drawn without lifting a pencil. It introduces discontinuous functions, highlighting cases where certain x values are not valid inputs, leading to undefined points or asymptotes. The tutorial also discusses discontinuities like holes and jumps, using examples to illustrate these concepts. Finally, it explains piecewise functions, which are evaluated differently based on the input value, and demonstrates how to graph them.

What characterizes a continuous function?

• It has gaps in its graph.

• It can be drawn without lifting the pencil.

• It is only defined for integer values.

• It has an asymptote.

Why is the function 1/(x-1) undefined at x=1?

• Because x=1 is not a real number.

• Because the numerator becomes zero.

• Because the denominator becomes zero.

• Because it is a piecewise function.

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

• It remains constant.

• It approaches negative infinity.

• It approaches positive infinity.

• It approaches zero.

In the function x^2-1/(x-1), what is the nature of the discontinuity at x=1?

• A hole in the graph.

• An asymptote.

• A continuous point.

• A jump discontinuity.

How is a piecewise function defined?

• By having no discontinuities.

• By being undefined for all x.

• By different expressions over different intervals.

• By a single expression for all x.