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What Is Continuity?
I’m sure that in your first pre-calculus class, you learned about the continuity of functions. If asked to define what this meant, most people would say, “A function is continuous if the graph of said function can be drawn without lifting the pencil or having any jumps.” Up until the late seventeenth century, this definition was more than sufficient. The Intermediate Value Property is one way to more precisely define continuity.
Definition 5.27: (Intermediate Value Property) Let f be defined on an interval I. Suppose that for each a, b ∈ I with f(a) 6= f(b), and for each d between f(a) and f(b), there exists c between a and b for which f(c) = d. We then say that f has the intermediate value property (IVP) on I. (Thompson et al. 305)
Example 5.28: Let F(x) = sin x−1 if x =/ 0
0 if x = 0.
The graph of F is shown below. You may wish to verify that F has the IVP. In particular, F
assumes every value in the interval [−1, 1] infinitely often in every neighborhood of x = 0. ◭
We haven’t yet made precise the phrase “the graph has no jumps,” but the IVP seems to convey that
idea well enough. Since this property is so easy to describe and appears to have content that is easy to
visualize, why not take it as the definition of continuity?
Before attempting to answer that question, let us offer a competing phrase to capture the idea of
continuity: “If x is near x0, then f(x) is near f(x0).” As stated, this phrase is not precise, but we can
make it precise using the limit concept. This phrase could be interpreted really as asserting that
f(x0) = lim f(x).
x→x0
According to this criterion our function F of Example 5.28 would not be continuous at x0 = 0, because
F(0) = 0, but limx→0 F(x) does not exist.
We shall see presently that the definition based on limits allows the development of a useful theory.
We’ll see that the class of continuous functions [as defined using equation (2)] is closed under addition
and multiplication, and that such functions have many other desirable properties. For example, the class
is closed under certain kinds of limits of sequences, and every continuous function on [a, b] is integratible.
On the other hand (as is shown in the exercises), none of the analogous statements is valid for the class of functions defined by IVP. Thus a theory of continuity based on the limit concept allows a rich structure and enjoys wide applicability, whereas one based on the IVP is rather limited. In addition, the fundamental notion of limit extends to much more general settings than R. In contrast, extensions of IVP, while possible, are peripheral to mathematical analysis. (Thompson et al. 306-307)
Continuity goes beyond just a overall generalization of the behavior of a function. It is also important that the points on a graph themselves are continuous.
Definition 5.29: (Continuous) Let f be defined in a neighborhood of x0. The function f is continuous at x0 provided limx→x0 f(x) = f(x0). (Thompson et al. 309)
This means that the function is continuous at a point x only if the limit of the function as it approaches that point is equal to the value of x itself. With this definition, it is possible for x to not be continuous in a couple of ways:
- If f is not defined at x
- limx→x0 f(x) does not exist
- F is defined at x and limx→x0 f(x) exists but
limx→x0 f(x) =/ f(x0) (Def. 5.29)
Examples (UCDavis p. 24):
Example 3.9. The sign function sgn : R → R, defined by sgn x = 1 if x > 0, 0 if x = 0, −1 if x < 0, is not continuous at 0 since limx→0 sgn x does not exist (see Example 2.6). The left and right limits of sgn at 0, lim x→0− f(x) = −1, lim x→0+ f(x) = 1, do exist, but they are unequal. We say that f has a jump discontinuity at 0
Example 3.10. The function f : R → R defined by f(x) = { 1/x if x ̸= 0, 0 if x = 0, is not continuous at 0 since limx→0 f(x) does not exist (see Example 2.7). Neither the left nor right limits of f at 0 exist either, and we say that f has an essential discontinuity at 0.
These definintions and examples relate to interior points in the function. There are also cases in which the points we speak of are not the interior points of the function but any arbitrary point or set.
We already know how to find the limit as our function approaches a point and use that information to see if the function is continuous at the point. What we don’t know yet however, is how to handle a situation in which our function has clearly defined endpoints. There is no way to check from both sides where the limit as our function approaches enpoints seeing as there is only one way to approach an end point. We change up our search for the limits as such:
lim x→a+ f(x) = f(a) and lim x→b− f(x) = f(b).
In this case, we’re approaching our endpoints from within the domain (a,b) in the real numbers.
https://math.stackexchange.com/questions/107296/can-continuity-be-proven-in-terms-of-closed-sets
This site has a nice discussion thread that goes through the proof of continuity using closed sets.
https://www.dpmms.cam.ac.uk/~wtg10/easyanalysis1.html
This site goes over the proof of of continuity using open sets
https://www.math.ucdavis.edu/~hunter/m125a/intro_analysis_ch3.pdf
Pages 21-24 give some more insight on the definitions related to continuity as well as a couple examples of its use.