Discontinuities

Front

What is a Jump Discontinuity?

Back

A Jump Discontinuity occurs when the left-hand limit and right-hand limit of a function at a certain point exist but are not equal, causing a 'jump' in the graph.

Front

What is a Removable Discontinuity?

Back

A Removable Discontinuity occurs when a function is not defined at a point, but the limit exists. It can be 'removed' by redefining the function at that point.

Front

What is an Infinite Discontinuity?

Back

An Infinite Discontinuity occurs when the function approaches infinity or negative infinity as it approaches a certain point, resulting in a vertical asymptote.

Front

What does it mean for a function to be Continuous at a point?

Back

A function is Continuous at a point if the limit as it approaches that point equals the function's value at that point.

Front

How can you identify a Jump Discontinuity on a graph?

Back

A Jump Discontinuity can be identified by observing a sudden 'jump' in the graph where the left-hand limit and right-hand limit do not match.

Front

What is the difference between Removable and Jump Discontinuities?

Back

Removable Discontinuities can be fixed by redefining the function at a point, while Jump Discontinuities cannot be fixed in this way as the limits do not match.

Front

What is the significance of limits in understanding discontinuities?

Back

Limits help determine the behavior of a function as it approaches a certain point, which is crucial for identifying types of discontinuities.

Front

Can a function have more than one type of discontinuity?

Back

Yes, a function can have multiple types of discontinuities at different points.

Front

What is an example of a function with a Jump Discontinuity?

Back

The piecewise function f(x) = { 1 for x < 0, 2 for x ≥ 0 } has a Jump Discontinuity at x = 0.

Front

What is an example of a function with a Removable Discontinuity?

Back

The function f(x) = (x^2 - 1)/(x - 1) has a Removable Discontinuity at x = 1, which can be fixed by defining f(1) = 2.