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:

xf(x)1123452345678holejumpasymptote
A very discontinuous function

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`qquad and qquad lim_(x->a)f(x) = f(a)`.

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

`f(x) = {(x^2 - 3,if x <= -1),(x - 1,if x > -1):}`.

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

`lim_(x->-1)f(x) = f(-1) = -2`,

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.

Continuity isn’t part of the grade twelve curriculum, so you won’t be tested on any of this. It won’t go away in university, though, so you might as well learn it now.