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 is continuous at if
is defined.
Let’s try using this definition. Consider the following piecewise function:
.
Is it continuous at −1? If it is, it must satisfy the two conditions in our definition. Since , the first condition is met. Since both pieces of are continuous individually (they are polynomials) and they intersect at , the left and right limits are equal and thus
,
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.