Continuity of a function is an easy concept to grasp intuitively, but the mathematical definition can be difficult to understand.  I will lead the reader in easy steps from the intuitive notion of continuity to its precise mathematical definition.

Although the development of this topic takes a while, the reward is that you will get to play the epsilon-delta math game at the end of the lesson.  If you have already read the introduction, and want to go directly to the game, click here.  Otherwise, brave reader, read on.
 

 function f                   function g

Notice that g has value 5 for x = 10, but then immediately jumps to value 3 as soon as x exceeds 10.  g is discontinuous for x = 10.

A function may be continuous at some points, and discontinuous at others (continuity is a point-wise property of a function).

The discussion does not end here.  The intuitive and informal definition we have given above is not precise enough for mathematicians.  Imprecision can lead to misleading thought and erroneous and/or contradictory conclusions.  I will show you what mathematicians have come up with (and only comparatively recently) as a more precise definition of continuity, one that can be used in mathematical proofs.

Here's g again:

Note that g(10) = 5, but for numbers x just a little larger than 10, no matter how close to 10, g(x) remains obstinately distant (by about 2) from g(10). Keeping this in mind, we may formulate continuity more precisely:

What does this mean? Let's consider f again:

Note that as x approaches 2, either from above or below, f(x) gets closer and closer to f(2) (=1), and stays close forever as x gets closer and closer to 10. We can make f(x) as close to f(2) as we want, by choosing x sufficiently close to 2.

On the other hand, this cannot be done for g at x = 10:

g(x) (for x a little more than 10) will never be arbitrarily close to g(10) , no matter how close x is to 10, because it obstinately stays at least two units away from g(10) (= 5).
 

How do mathematicians express (symbolically) the idea that number x is "close to" number y?  They say that the "distance" between x and y can be made "arbitrarily small".  And they use the symbols e (Greek letter "epsilon") and d ("delta") to stand for "arbitrarily small quantities", as in the following

We can also put the statement "the distance of x from x₀ is smaller than d " into mathematical language. Recall that

represents the (positive) distance of x from a.  So "the distance of x from x₀ is smaller than d " can be written as

and similarly for "the distance of f(x) from f(x₀)" .  Hence, we come up with a truly mathematical definition of continuity (at a point):
 

Here's the same definition, but stated more concisely by using mathematical notation more extensively:

Of course, an underlying assumption is that the function under consideration is defined for all values x in some neighborhood of x₀. If not, the function will be discontinuous at that point right off the bat.  Think about the following pathological function p: p(x) = 1 for all rational values of x, and is undefined for all irrational values. Is p continuous anywhere?
 

Let's explore the e - d definition a little more by examining the following function f:

The point to be considered is x₀ = 2, for which f(x₀) = .5.  The definition says "Given any e>0 … " (click here to review it).  For example, let's pick e =.3. We will depict the e by drawing a (2 x e = .6) wide band around f(2) = .5:

Our job is to pick d so that for all x within d of 2, f(x) is within .3 of f(2), that is, within the e band. As our first try, we choose d = .4:

Notice that all f(x) values within the e band are rendered in red, but that there are values of f(x) within the d band that are blue, that is, outside the e band, and hence, not within e of f(2). So this particular d does not satisfy the criterion.

Is there one that does? (We need find only one!).  Let's reduce the size of d to .1 (you can probably see in advance that this will work, but here's what it looks like):

Now the function values within the d-band are all red, meaning that for the given e, we have found a d such that for all x within d of 2, f(x) is within e of .5. So, for the given e, we have found the required d, and we are done.

This, of course does not prove the continuity of f at x = 2, for two reasons:

1. Our argument depends on what we see on the graph of f.  Mathematical proof cannot rely on pictures, as they can be misleading.
2. We have conducted the demonstration only for e = .5.  The definition requires that it be demonstrated for all e.

Now we come to a demonstration of the "search" for an appropriate d given an e. The problem is this:

We show you the computer doing the search, by starting out at d = .3, and narrowing it until we get one that works. Click in the empty space below to see it. Click on the picture to see it again. The status window at the bottom of the frame tells you what's happening.

What follows is a little epsilon-delta game you can play. There are two styles of play:

1. You supply the epsilon, and the computer will find a delta.
2. The computer chooses an epsilon, you choose a delta. You will see the results.

To go back to the beginning, click here.