Asymptotes and discontinuities

Asymptotes

An asymptote of a function is a straight line which the function values approach under certain conditions. There are three types of asymptotes: horizontal, vertical and oblique.

Horizontal asymptotes have equations of the form $y=c$y=c. They occur when the function approaches a constant value $c$c as $x$x tends towards positive or negative infinity.

Vertical asymptotes have equations of the form $x=c$x=c. They occur when the function values tend towards positive or negative infinity as $x$x approaches the constant value $c$c.

It is also possible for a function to have an oblique asymptote or slanted asymptote. Graphically, this means that the function approaches a straight line with a non-zero slope as $x$x tends towards positive or negative infinity.

As an example, here is the graph of a hyperbolic function:

Notice that the function values tend towards $\pm\infty$±∞ as the values of $x$x approach $1$1. This function has a vertical asymptote at $x=1$x=1, which has been displayed as a dashed line.

There is also a horizontal dashed line with equation $y=2$y=2, which is the horizontal asymptote of this function. We can see that the function values approach $2$2 as the values of $x$x tend towards positive and negative infinity.

 

 

Continuity

Recall that the domain of a function is the set of $x$x-values on which the function is defined. Graphically, we can think of the domain as all values of $x$x which correspond to a point on the curve.

A function $f\left(x\right)$f(x) is said to be continuous at a point $x=c$x=c if:

  i. the function is defined at that point (that is, $c$c is in the domain of $f\left(x\right)$f(x)), and

  ii. we can get function values as close as we like to $f\left(c\right)$f(c) by taking a small enough region of domain values around $x=c$x=c.

A function $f\left(x\right)$f(x) is then said to be a continuous function if it is continuous at all points in its domain. Similarly, a function is continuous over an interval if it is continuous at all points in that interval.

If instead there exists a point in the function's domain at which it is not continuous, we say that the function is discontinuous at that point.

 

Continuous functions

Linear functions (that is, functions of the form $f\left(x\right)=ax+b$f(x)=ax+b) are defined for all real numbers. We can represent this by saying its domain is the interval $\left(-\infty,\infty\right)$(−∞,∞).

Look at the linear function in the graph above. At the domain value $x=2$x=2 we have the function value $f\left(2\right)=3$f(2)=3. We can get other function values as close as we like to $3$3 by taking a small enough region of the domain around $x=2$x=2. Watch the applet below for a demonstration.

Created with Geogebra

There is nothing special about $x=2$x=2 in this case - the same is true for any other point in the domain! So the function above is a continuous function, as are all polynomial functions.

 

Discontinuous functions

Most of the functions that we are already familiar with are continuous. Not all functions are continuous though! Let's take a look at an example of a step function:

In the graph above, the step function has a step at $x=0$x=0. It takes a constant value (in this case $-1$−1) for $x<0$x<0 and a different constant value (in this case $+1$+1) for $x\ge0$x≥0, and so its domain is $\left(-\infty,\infty\right)$(−∞,∞) (that is, all of the real numbers).

This function has a point of discontinuity at $x=0$x=0 - no matter how small we make our region of domain values around $x=0$x=0, we will always include part of the function that takes the value $-1$−1 and part of the function that takes the value $+1$+1. This means that the step function is not a continuous function.

 

 

Practice questions

Question 1

Question 2