Determining Whether a Function is Continuous

Date: 06/10/2000 at 22:05:02
From: John Keenan
Subject: Continuous functions

How can you tell whether or not a function is continuous?

Date: 06/13/2000 at 18:11:10
From: Doctor Maureen
Subject: Re: Continuous functions

Dear John,

Welcome to the Math Forum and thanks for your interesting question. 
You can approach this question graphically or algebraically (that is, 
by analyzing the function without looking at its graph).

I'll address continuity from a graphical standpoint first. A function 
is continuous if you can sketch the entire graph without lifting your 
pencil from the paper. In other words, the graph has no breaks in it. 
One example is the graph of a parabola, f(x) = x^2 + 1. The graph for 
this function is continuous because you can plot a y-value for every 
possible x-value, and the y-values have no "sudden jumps," so the 
graph is a smooth continuous graph. An example of a discontinuous 
graph is g(x) = 1/x. Since the variable is in the denominator, g(x) is 
not defined for x = 0. When you look at the graph for this function, 
you will see an asymptote at x = 0, which is the y-axis. When you try 
to sketch this graph, you may plot a few points and then attempt to 
connect them into one curve. However, as you approach zero from the 
negative direction, you will have to stop at x = 0, lift your pencil 
and start again at the lowest positive number.

You might be wondering, what if I can't see the graph of the function 
- how will I know if it is continuous? Well there are really only two 
kinds of functions that you will have to analyze for continuity, 
rational functions in which there is a fraction and the variable is in 
the denominator, and piecewise functions. Here are some examples of 
rational functions:

     f(x) = 2/(x-1)
     g(x) = (x+3)/(x^2-4)
     h(x) = 5/(x^3+1)

In each of these functions, x is in the denominator and there is a 
value or values for x that will make the function undefined. Do you 
see that for h(x)? If x = -1, the denominator will be 0. Therefore, 
h is discontinuous at -1. For what values of x are f(x) and g(x) 
discontinuous? In these cases the points of discontinuity are directly 
related to the domain. For h(x), the domain is all real numbers except 
-1. For f(x), the domain is all real numbers except 1. So the 
functions are discontinuous at the points that are not defined in the 
domain. 

This is not always the case. Consider the function

     f(x) = sqrt(x-5)

(Sqrt stands for square root - I can't type the radical symbol into 
this e-mail.) In this case, f is defined for all real numbers greater 
than or equal to 5 since you can't take the square root of a negative 
number. Even though the domain of f is limited, f is considered 
continuous along its defined values. Its curve starts at (5,0) and 
goes on to infinity without a break in the graph. It is only rational 
functions that have a direct relationship between the points of 
discontinuity and their domain.

The other type of function is the piecewise function, in which a 
function has two or more definitions depending of the value of x. For 
example, consider g(x) = x-1 if x < 0, and g(x) = x^2 if x >= 0. Can 
you picture that g(x) is a straight line from negative infinity until 
x = 0? Then at 0, g turns into a parabola for the positive x values. 
At x = 0, the straight-line portion of the graph will end at the point 
(0,1), and at (0,0) the parabola will start. So there will be break in 
the curve; your pencil will not be able to continue the curve without 
lifting at (0,1) and sketching part of a parabola again at (0,0). When 
analyzing a piecewise function, you must determine what happens at the 
point(s) where the definition changes. Sometimes the two definitions 
overlap and there is a continuous graph, other times there is a gap 
between the two curves. In the example I just described, do you see 
that if I changed the second definition for g to g(x) = x^2 + 1 then 
the two graphs would overlap at (0,1)? Thus, g would be continuous.  

My explanation got a bit lengthy. To summarize, continuous functions 
are functions in which the graphs are smooth curves or lines without 
breaks in them. The two types of functions that may have points of 
discontinuity are rational and piecewise functions. Rational functions 
will be discontinuous at the points where the function is undefined. 
Piecewise functions may be discontinuous at the points where the 
function definition changes. In the latter, you will see a gap in the 
graph.

I hope this explanation helps. Please write back if I used language 
that is unfamiliar to you.

- Doctor Maureen, The Math Forum
  http://mathforum.org/dr.math/