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x’!a - a
x’!a
x
4.2. LIMITS AND CONTINUITY 31
We illustrate the behaviour of the function for the case when a = 2 in Fig 4.1
6
5
4
3
2
1
-1
-2
-4 -3 -2 -1 0 1 2 3 4
Figure 4.1: Graph of the function (x2 - 4)/(x - 2) The automatic graphing routine does
not even notice the singularity at x =2.
In this example, we can argue that the use of the (x2 - a2)/(x - a) was perverse; there
was a more natural definition of the function which gave the right answer. But in the
case of sin x/x, example 4, there was no such definition; we are forced to make the two part
definition, in order to define the function properly everywhere. So we again have to be
careful near a particular point in this case, near x = 0. The function is graphed in Fig 4.2,
and again we see that the graph shows no evidence of a difficulty at x =0
Considering example 5 shows that these limits need not always exist; we describe this
by saying that the limit from the left and from the right both exist, but differ, and the
function has a jump discontinuity at 0. We sketch the function in Fig 4.3.
In fact this is not the worst that can happen, as can be seen by considering example 6.
Sketching the graph, we note that the limit at 0 does not even exists. We prove this in
more detail later in 4.23.
The crucial property we have been studying, that of having a definition at a point which
is the right definition, given how the function behaves near the point, is the property of
continuity. It is closely connected with the existence of limits, which have an accurate
definition, very like the sequence ones, and with very similar properties.
4.3. Definition. Say that f(x) tends to l as x ’! a iff given >0, there is some ´ >0
such that whenever 0
Note that we exclude the possibility that x = a when we consider a limit; we are only
interested in the behaviour of f near a, but not at a. In fact this is very similar to the
definition we used for sequences. Our main interest in this definition is that we can now
describe continuity accurately.
4.4. Definition. Say that f is continuous at a if limx’!a f(x) =f(a). Equivalently, f
is continuous at a iff given >0, there is some ´ >0 such that whenever |x - a|
|f(x) - f(a)|
32 CHAPTER 4. LIMITS AND CONTINUITY
1
0.8
0.6
0.4
0.2
-0.2
-0.4
-8 -6 -4 -2 0 2 4 6 8
Figure 4.2: Graph of the function sin(x)/x. Again the automatic graphing routine does
not even notice the singularity at x =0.
y
x
Figure 4.3: The function which is 0 when x
discontinuity at x =0.
Note that in the epsilon - delta definition, we no longer need exclude the case when
x = a. Note also there is a good geometrical meaning to the definition. Given an error
, there is some neighbourhood of a such that if we stay in that neighbourhood, then f is
trapped within of its value f(a).
We shall not insist on this definition in the same way that the definition of the con-
vergence of a sequence was emphasised. However, all our work on limts and continuity of
functions can be traced back to this definition, just as in our work on sequences, everything
could be traced back to the definition of a convergent sequence. Rather than do this, we
shall state without proof a number of results which enable continuous functions both to be
recognised and manipulated. So you are expected to know the definition, and a few simply
´ proofs, but you can apply (correctly - and always after checking that any needed
conditions are satisfied) the standard results we are about to quote in order to do required
manipulations.
4.5. Definition. Say that f : U(open) ’! R is continuous if it is continuous at each point
4.2. LIMITS AND CONTINUITY 33
1
0.5
-0.5
-1
0 0.05 0.1 0.15 0.2 0.25 0.3
Figure 4.4: Graph of the function sin(1/x). Here it is easy to see the problem at x =0;
the plotting routine gives up near this singularity.
a " U.
Note: This is important. The function f(x) = 1/x is defined on {x : x = 0}, and
is a continuous function. We cannot usefully define it on a larger domain, and so, by the
definition, it is continuous. This is an example where the naive can draw it without taking
the pencil from the paper definition of continuity is not helpful.
x3 -8
4.6. Example. Let f(x) = for x = 2. Show how to define f(2) in order to make f
x -2
a continuous function at 2.
Solution. We have [ Pobierz całość w formacie PDF ]
