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.

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