Continuity Rules

In order for f(x) to be continuous at point x = c, all three of these conditions must be true:

- f(c) exists - lim(x->c) f(x) exists - lim(x->c) f(x) = f(c)

What is a jump discontinuity?

A jump discontinuity occurs when the curve "breaks" at a particular place and starts somewhere else. In other words, lim(x->a-) f(x) ≠ lim(x->a+) f(x).

What is a point discontinuity?

A point discontinuity occurs when the curve has a "hole" in it from a missing point because the function has a value at that point that is "off the curve". In other words, lim(x->a) f(x) ≠ f(a).

What is a removable discontinuity?

A removable discontinuity occurs when you have a rational expression with common factors in the numerator and denominator. Because these factors can be canceled, the discontinuity can be "removed".

What is an essential/infinite discontinuity?

An essential/infinite discontinuity occurs when the curve has a vertical asymptote.

What does the Intermediate Value Theorem state?

If a function is continuous on the interval [a, b] (inclusive), then for any value c in the interval (a, b) (not inclusive), there exists a value f(c) between f(a) and f(b) (inclusive)

What are the four types of discontinuities?

Jump, point, essential (also called infinite), and removable