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  • § 4.2 - Discontinuity of a function
    • Discontinuity
    • Extensions of functions
    • Types of discontinuities

(last updated: 1:23:48 PM, November 03, 2020)

§ 4.2 - Discontinuity of a function

Discontinuity

This next definition appears in my 4.1 notes already.

Definition (not in text, but consistent with text’s terminology usage) A function f is said to be discontinuous at a if

  • a∈D and f is not continuous at a, or
  • a∉D. Note: on this point, I disagree with the textbook terminology. I prefer that when a function is not defined at a to say the function cannot be continuous or discontinuous there. The function does nothing at a and therefor cannot be anything there.

Although I depart from the textbooks terminology standards here, it is simply a matter of choice of language, a personal preference and is of no consequence.

Extensions of functions

Definition 4.2.1 A function $g : ER R $ is an extension of the function $f : DR R $ provided that D⊂E and f(x)=g(x) for all x∈D. If g is continuous, then g is called a continuous extension of f.

Types of discontinuities

The textbook defines several different types of discontinuities.

  • Removable discontinuity (Definition 4.2.3). This is a “hole” (or swath of holes) in the graph of the function that could be filled (or a singleton point that can be moved/redefined) so as to make the function continuous there.
  • Jump discontinuity (Definition 4.2.5). This is a point where the function jumps/skips over an interval of y-values.
  • Infinite discontinuity (Definition 4.2.6). This is point where there is a vertical asymptote.

There are further ways to classify other discontinuities as well. You should read these definitions, but you will not be tested on them.

Note that I prefer to only say the function is discontinuous at x=a when a is actually a point in the domain of f. When a∉Dom(f) I say f is not continuous there since it is not defined there in which case it might be possible to extend the function and define it there continuously.

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