1 Continuous and discontinuous functions and limits

Look at the graph shown in Figure 18a. The curve can be traced out from left to right without moving the pen from the paper. The function represented by this curve is said to be continuous at every point. If we try to trace out the curve in Figure 18b, the presence of a jump in the graph (at $x=x_1$ ) means that the pen must be lifted from the paper and moved in order to trace the graph. Such a function is said to be discontinuous at the point where the jump occurs. The jumps are known as discontinuities.

Task!

Sketch a graph of a function which has two discontinuities.

When defining a discontinuous function algebraically it is often necessary to give different function rules for different values of $x$ . Consider, for example, the function defined as:

$f\left(x\right)=\left\{\begin{array}{cc}3\hfill & x<0\hfill \\ x^2\hfill & x\ge 0\hfill \\ \hfill \end{array}\right.$

Notice that there is one rule for when $x$ is less than 0 and another rule for when $x$ is greater than or equal to 0.

A graph of this function is shown in Figure 19.

Figure 19 :

Suppose we ask ‘to what value does $y$ approach as $x$ approaches 0?’. From the graph we see that as $x$ gets nearer and nearer to 0, the value of $y$ gets nearer to 0, if we approach from the right-hand side. We write this formally as

$\underset{x\to 0^{+}}{lim}f\left(x\right)=0$

and say ‘the limit of $f\left(x\right)$ as $x$ tends to 0 from above is 0.’

On the other hand if $x$ gets closer to zero, from the left-hand side, the value of $y$ remains at 3. In this case we write

$\underset{x\to 0^{-}}{lim}f\left(x\right)=3$

and say ‘the limit of $f\left(x\right)$ as $x$ tends to 0 from below is 3.’

In this example the right-hand limit and the left-hand limit are not equal, and this is indicative of the fact that the function is discontinuous.

In general a function is continuous at a point $x=a$ if the left-hand and right-hand limits are the same there and are finite, and if both of these are equal to the value of the function at that point. That is

If the right-hand and left-hand limits are the same, we can simply describe this common limit as $\underset{x\to a}{lim}f\left(x\right)$ . If the limits are not the same we say the limit of the function does not exist at $x=a$ .

Exercises

1. Explain the distinction between a continuous and a discontinous function. Draw a graph showing an example of each type of function.
2. Study graphs of the functions $y=x^2$ and $y=-x^2$ . Are these continuous functions?
3. Study graphs of $y=3x-2$ and $y=-7x+1$ . Are these continuous functions?
4. Draw a graph of the function

   $f\left(x\right)=\left\{\begin{array}{cc}2x+1\hfill & x<3\hfill \\ 5\hfill & x=3\hfill \\ 6\hfill & x>3\hfill \end{array}\right.$

   Find

   1. $\underset{x\to 0^{+}}{lim}f\left(x\right)$ ,
   2. $\underset{x\to 0^{-}}{lim}f\left(x\right)$ ,
   3. $\underset{x\to 0}{lim}f\left(x\right)$ ,
   4. $\underset{x\to 3^{+}}{lim}f\left(x\right)$ ,
   5. $\underset{x\to 3^{-}}{lim}f\left(x\right)$ ,
   6. $\underset{x\to 3}{lim}f\left(x\right)$ ,

Answer