## Content

### Continuity

When first showing students the graph of $$y=x^2$$, we generally calculate the squares of a number of $$x$$-values and plot the ordered pairs $$(x,y)$$ to get the basic shape of the curve. We then join the dots' to produce a connected curve.

We can justify this either by plotting intermediary points to show that our plot is reasonable or by using technology to plot the graph. That we can join the dots' is really the consequence of the mathematical notion of continuity.

A formal definition of continuity is not usually covered in secondary school mathematics. For most students, a sufficient understanding of continuity will simply be that they can draw the graph of a continuous function without taking their pen off the page. So, in particular, for a function to be continuous at a point $$a$$, it must be defined at that point.

Almost all of the functions encountered in secondary school are continuous everywhere, unless they have a good reason not to be. For example, the function $$f(x) = \dfrac{1}{x}$$ is continuous everywhere, except at the point $$x=0$$, where the function is not defined.

A point at which a given function is not continuous is called a discontinuity of that function.

Here are more examples of functions that are continuous everywhere they are defined:

• polynomials (for instance, $$3x^2 + 2x - 1$$)
• the trigonometric functions $$\sin x$$, $$\cos x$$ and $$\tan x$$
• the exponential function $$a^x$$ and logarithmic function $$\log_b x$$ (for any bases $$a > 0$$ and $$b > 1$$).

Starting from two such functions, we can build a more complicated function by either adding, subtracting, multiplying, dividing or composing them: the new function will also be continuous everywhere it is defined.

#### Example

Where is the function $$f(x)= \dfrac{1}{x^2-16}$$ continuous?

#### Solution

The function $$f(x) = \dfrac{1}{x^2-16}$$ is a quotient of two polynomials. So this function is continuous everywhere, except at the points $$x=4$$ and $$x=-4$$, where it is not defined.

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