Limits, Continuity, IVT

Front

What is a finite jump discontinuity?

Back

A finite jump discontinuity occurs when the left-hand limit and right-hand limit of a function at a point exist but are not equal, resulting in 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 the input approaches a certain value, leading to a vertical asymptote.

Front

What is a removable discontinuity?

Back

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

Front

State the Intermediate Value Theorem (IVT).

Back

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b], and N is any number between f(a) and f(b), then there exists at least one c in (a, b) such that f(c) = N.

Front

How do you determine if a function is continuous at a point?

Back

A function f is continuous at a point a if: 1) f(a) is defined, 2) the limit of f(x) as x approaches a exists, and 3) the limit equals f(a).

Front

What does it mean for a function to be continuous on an interval?

Back

A function is continuous on an interval if it is continuous at every point within that interval.

Front

Provide an example of a function with a finite jump discontinuity.

Back

The piecewise function f(x) = { 1 for x < 0, 2 for x = 0, 3 for x > 0 } has a finite jump discontinuity at x = 0.

Front

Provide an example of a function with an infinite discontinuity.

Back

The function f(x) = 1/(x-1) has an infinite discontinuity at x = 1.

Front

What is the significance of the limits in determining continuity?

Back

Limits help determine continuity by showing whether the function approaches a specific value as the input approaches a point.

Front

How can you identify intervals where a polynomial function is positive?

Back

To find intervals where a polynomial function is positive, identify the roots and test the sign of the function in the intervals defined by these roots.