# What is discontinuity in calculus?

Jul 16, 2015

I would say that a function is discontinuous at $a$ if it is continuous near $a$ (in an open interval containing $a$), but not at $a$. But there are other definitions in use.

#### Explanation:

Function $f$ is continuous at number $a$ if and only if:

${\lim}_{x \rightarrow a} f \left(x\right) = f \left(a\right)$

This requires that:
1 $\text{ }$ $f \left(a\right)$ must exist. ($a$ is in the domain of $f$)

2 $\text{ }$ ${\lim}_{x \rightarrow a} f \left(x\right)$ must exist

3 The numbers in 1 and 2 must be equal.

In the most general sense: If $f$ is not continuous at $a$, then $f$ is discontinuous at $a$.

Some will then say that $f$ is discontinuous at $a$ if $f$ is not continuous at $a$

Others will use "discontinuous" to mean something different from "not continuous"

One possible additional requirement is that $f$ be defined "near" $a$ -- that is: in an open interval containing $a$, but perhaps not at $a$ itself.

In this usage, we would not say that $\sqrt{x}$ is discontinuous at $- 1$. It is not continuous there, but "discontinuous" requires more.

A second possible additional requirement is that $f$ must be continuous "near" $a$.
In this usage:
For example: $f \left(x\right) = \frac{1}{x}$ is discontinuous at $0$,

But $g \left(x\right) = \left\{\begin{matrix}0 & \text{if" & x & "is rational" \\ 1 & "if" & x & "is irrational}\end{matrix}\right.$

which is not continuous for any $a$, has no discontinuities.

A third possible requirement is that $a$ must be in the domain of $f$ (Otherwise, the term "singularity" is used.)

In this usage $\frac{1}{x}$ in not continuous at $0$, but it is also not discontinuous because $0$ is not in the domain of $\frac{1}{x}$.

My best advice is to ask the person who will be evaluating your work which usage they prefer. And otherwise, don't worry too much about it. Be aware that there are various ways of using the word and they are not all in agreement.