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Grade 11

Graphing continuous and discontinuous functions

Lesson

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.

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.

 

Remember!

If the graph of a function has a vertical 'jump' at a point in its domain, then we say that the function is discontinuous. Otherwise, we say that the function is continuous over its domain.

 

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.

 

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.

Careful!

If a function is smooth then it is continuous over its domain. However not all continuous functions are smooth.

 

Worked example

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

   

$x+2$x+2


$2$2

when $x\ge0$x0


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 $x\ge0$x0. 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

Consider the graph of the function $f\left(x\right)$f(x) $=$= $-4$4 when $x<2$x<2
$4$4 when $x\ge2$x2

Loading Graph...

  1. Are there any points within its domain where the function is discontinuous?

    Yes

    A

    No

    B
  2. As such, is the function continuous over its domain?

    Yes

    A

    No

    B

Outcomes

11U.C.1.1

Make connections between sequences and discrete functions, represent sequences using function notation, and distinguish between a discrete function and a continuous function

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