The Concept of Continuity and Basic Properties

4.1 The Concept of Continuity and Basic Properties

Structures are the weapons of the mathematician.

Bourbaki Intuitively, a function is continuous on an interval if its graph can be drawn “con-

tinuously,” that is, without lifting the pen from the paper. Rigorously, a function f defined on an interval (a, b) is continuous at some point c ∈ (a,b) if for each ε > 0, there exists δ > 0 depending on both ε and c such that | f (x) − f (c)| < ε whenever |x − c| < δ . Equivalently, f is continuous at c if and only if it has left-hand and right-hand limits at c and they are equal to each other and to f (c), that is,

lim f (x) = lim

x → c + f (x) = f (c),

or

lim f (x) = f (c). x → c

In other words, f is continuous at c if f commutes with “lim x → c ,” namely

lim f (x) = f lim x

140 4 Continuity This definition is equivalent to the following sequential version: a function f

defined on an interval (a, b) is continuous at c ∈ (a,b) if lim n → ∞ f (x n ) = f (c) for any sequence (x n ) n ≥1 ⊂ I such that lim n → ∞ x n = c.

The set of all functions f : I →R that are continuous on I is denoted by C(I). To simplify the notation, we shall often write C (a, b) and C[a, b] instead C((a, b)) and

C ([a, b]), respectively. Continuity is a local property (that is, it depends on any point), but it can be extended to a whole set. More precisely, if f : I →R is a function and A ⊂ I, then we say that f is continuous on A if it is continuous at every point a ∈ A. In such a case we have obtained a global property.

If f and g are functions that are continuous on an interval I, then f + g, f − g, and f g are continuous on I, and f

g is continuous on I and f is continuous on J, where g is an interval that contains the range of g, then f ◦ g is continuous on I. Continuity may be also defined by means of open sets. More precisely, we have

f ∈ C(I) if for any open set V ⊂ R, there exists an open set U ⊂ R such that

f −1 (V ) = I ∩U. In particular, if I is open, then f is continuous on I if the inverse image under f of every open set is also an open set. The same property can be stated in terms of closed sets.

The direct image of an open (resp, closed) set under a continuous function is not necessarily an open (resp., closed) set. However, the image of a compact (that is, closed and bounded) set under a continuous function is also compact. In particular, if f is a continuous function with compact domain K, then there exists a positive constant C such that | f (x)| ≤ C for all x ∈ K. A more precise property is stated in the following result, which is called “Hauptlehrsatz” (that is, principal theorem) in Weierstrass’s lectures of 1861.

Weierstrass’s Theorem. Every real-valued continuous function on a closed and bounded interval attains its maximum and its minimum.

Let f be a function with domain I. Let a ∈ I and assume that f is discontinuous at a. Then there are two ways in which this discontinuity can occur:

Fig. 4.1 Discontinuity of the first kind.

4.1 The Concept of Continuity and Basic Properties 141

Fig. 4.2 Discontinuity of the second kind (limit does not exist).

(i) If lim x → a − f (x) and lim x → a + f (x) exist, but either do not equal each other or do not equal f (a), then we say that f has a discontinuity of the first kind at the point a (see Figure 4.1).

(ii) If either lim x → a − f (x) does not exist or lim x → a + f (x) does not exist, then we say that f has a discontinuity of the second kind at the point a. We illustrate this notion in Figures 4.2 and 4.3, which correspond to the functions

Fig. 4.3 Discontinuity of the second kind (infinite limit).

In the first case, sin (1/x) produces an infinity of oscillations in a neighborhood of the origin as x tends to zero. However, the function

is continuous (see Figure 4.4). As mentioned by Weierstrass (1874), “Here the oscillations close to the origin are less violent, due to the factor x, but they are still infinitely small.”

Examples. Consider the functions

1 if x is rational ,

f 3 (x) =

0 if x is irrational.

Then f 1 has a discontinuity of the first kind at 0 , while f 2 has a discontinuity of the second kind at the origin. The function f 3 has a discontinuity of the second kind at every point.

Fig. 4.4 Graph of the function f

4.1 The Concept of Continuity and Basic Properties 143 Riemann (1854) found the following example of a function that is discontinuous

in every interval:

∞ B (nx)

f (x) = ∑

, with B (x) =

n =1 n 2 0 if x = k/2, where k

at x = 1/2, 1/4, 3/4, 1/6, 3/6, 5/6, . . . . If f is a continuous, strictly monotone function with domain [a, b], then its inverse

f −1 exists and is continuous.

A refined type of continuity is now described. If f is a function with domain D, then f is called uniformly continuous on D if for any ε > 0, there exists δ > 0 such that | f (x) − f (y)| < ε whenever x, y ∈ D and |x − y| < δ . We observe that in this definition δ is “uniform” with respect to x, y ∈ D, in the sense that δ depends only on ε and is independent of x and y. The main property related to uniform continuity is that if f is a continuous function with compact domain K, then f is uniformly continuous on K.

A function f : I →R defined on an interval I is said to have the intermediate value property if for all a and b in I with a < b and for any number y between f (a) and

f (b), there exists a number x in [a, b] such that f (x) = y.

A basic property of continuous functions defined on an interval is that they have the intermediate value property. This theorem appears geometrically evident and was used by Euler and Gauss. Only Bolzano found that a “rein analytischer Beweis” was necessary to establish more rigor in analysis. It was widely believed by many mathematicians in the nineteenth century that the intermediate value property is equivalent to continuity. The French mathematician Gaston Darboux (1842–1917) proved in 1875 that this is not the case (see [20]). We will discuss in detail in the next chapter a celebrated result of Darboux that asserts that any derivative has the intermediate value property. Furthermore, Darboux gave examples of differentiable functions with discontinuous derivatives. In conclusion, for real-valued functions defined on intervals,

Continuity =⇒ Intermediate Value Property,

but the converse is not true.

Exercise. Prove that the function f :R →R defined by

has the intermediate value property if and only if −1 ≤ a ≤ 1 . This result shows that a function that does not have the intermediate value property may be written as the sum of two functions that do have this property.

144 4 Continuity Indeed, if

then f =f 1 +f 2 , where

A deep result due to the Polish mathematician Wacław Sierpi´nski (1882–1969) asserts that if I is an interval of real numbers, then any function f : I →R can be written as f =f 1 +f 2 , where f 1 and f 2 have the intermediate value property.

A natural question related to the previous results is the following. If f and g are functions defined on [a, b] such that f is continuous and g has the intermediate value property, does f + g have the intermediate value property? Most persons would be inclined to answer “yes.” Nevertheless, the correct answer is “no,” and we refer to [80] for more details and a related (but complicated!) counterexample.