Graphing continuous and discontinuous functions

A function $f\left(x\right)$f(x) is said to be continuous at a point in its domain $x=c$x=c if we can get as close as we like to the point $f\left(c\right)$f(c) by taking a small enough region of domain values around $x=c$x=c. Take a look at the applet below.

Created with Geogebra

This demonstrates that there are no sharp changes in the function values around $f\left(c\right)$f(c). If we can do this for all the points in the domain, we say that the function is continuous over its domain.

 

Functions that are discontinuous do not satisfy this property, and will have some sharp change in the function values around $f\left(c\right)$f(c). Notice that in the next applet, domain values close to $x=2$x=2 do not correspond to function values close to $f\left(2\right)=5$f(2)=5. So we say that this function is discontinuous.

Created with Geogebra

 

 

We can classify continuous functions even further into the categories of smooth and not smooth functions. We say a function is smooth if we can always draw a tangent line against any point in its domain. For example, the parabola in the applet below is smooth since a tangent line can be drawn at any point in its domain.

Created with Geogebra

 

Functions that are not smooth form sharp points at one or more parts of the graph, where tangent lines can no longer be defined. For example, the function $f\left(x\right)=\left|x\right|$f(x)=|x| is clearly not smooth since the tangent line at the point $x=0$x=0 is undefined.

$f\left(x\right)=\left|x\right|$f(x)=|x| is not smooth

 

The function $f\left(x\right)=\left|x\right|$f(x)=|x| is continuous but not smooth. This leads us to the following important claim.

 

Worked example

Draw the graph of the following function and state whether it's continuous and smooth.

			$x+2$x+2 $2$2	when $x\ge0$x≥0 when $x<0$x<0
$f\left(x\right)$f(x)	$=$=

Think: The graph of the function describes the pieces of two distinct lines. The first describes the line $y=x+2$y=x+2 but restricted to the domain $x\ge0$x≥0. The other describes the horizontal line $y=2$y=2 but restricted to the domain $x<0$x<0.

Do: Drawing each of the pieces, we get the following graph of $f\left(x\right)$f(x):

Graph of the function $y=f\left(x\right)$y=f(x)

 

Since the graph of the function has no 'jumps' in its domain, the function is continuous over its domain. However, the function is not smooth since it contains a sharp point at $x=0$x=0.

 

Practice question