Continuity

A continuous function is one that can be drawn without lifting your pencil. A discontinuous function cannot be drawn like this because it has one or more discontinuities. There are three types of discontinuities:

• point discontinuity (hole)
• jump discontinuity (jump)
• infinite discontinuity (asymptote)

We can also talk about whether a function is continuous at a particular point. For example, the function graphed above is continuous at 3.

Drawing a curve without lifting your pencil is a good way of thinking about continuity, but this isn’t a very good definition. Here is a formal definition with some mathematical rigour in it: a function $f$ is continuous at $a$ if

$f(a)$ is defined and $\underset{x\to a}{\lim }f(x)=f(a)\text{.}$

Let’s try using this definition. Consider the following piecewise function:

$f(x)=\{\begin{array}{ll}{x}^2-3& \text{if}x\le -1\\ x-1& \text{if}x>-1\end{array}\text{.}$

Is it continuous at −1? If it is, it must satisfy the two conditions in our definition. Since $f(-1)=-2\text{,}$ the first condition is met. Since both pieces of $f$ are continuous individually (they are polynomials) and they intersect at $x=-1\text{,}$ the left and right limits are equal and thus

$\underset{x\to -1}{\lim }f(x)=f(-1)=-2\text{,}$

and the second condition is met. If this doesn’t make sense, try graphing the function. You will see that the function approaches −2 from both the left and the right.