Discontinuities in Functions Start

Front

What is a discontinuity in a function?

Back

A discontinuity in a function occurs when there is a break, jump, or hole in the graph of the function, meaning the function is not continuous at that point.

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 of the function as it approaches that point equals the function's value at that point.

Front

What is the limit of a polynomial function as x approaches c?

Back

Front

What is the limit of a rational function as x approaches c?

Back

Front

What are the types of discontinuities in functions?

Back

There are three main types: removable discontinuities (holes), jump discontinuities, and infinite discontinuities (asymptotes).

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 often be 'fixed' by redefining the function at that point.

Front

What is a jump discontinuity?

Back

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

Front

What is an infinite discontinuity?

Back

An infinite discontinuity occurs when the function approaches infinity (or negative infinity) as x approaches a certain value, typically seen in vertical asymptotes.

Front

How can you identify a discontinuity in a function's graph?

Back

Look for breaks, jumps, holes, or vertical asymptotes in the graph of the function.

Front

What is the significance of limits in understanding discontinuities?

Back

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