Page 56 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 3] FUNCTIONS, LIMITS, AND CONTINUITY 47

2
x ; x 6¼ 2
EXAMPLES. 1. If f ðxÞ¼ then from the example on Page 45 lim f ðxÞ¼ 4. But f ð2Þ¼ 0. Hence
0; x ¼ 2 x!2
lim f ðxÞ 6¼ f ð2Þ and the function is not continuous at x ¼ 2.
x!2
2
2. If f ðxÞ¼ x for all x,then lim f ðxÞ¼ f ð2Þ¼ 4 and f ðxÞ is continuous at x ¼ 2.
x!2
Points where f fails to be continuous are called discontinuities of f and f is said to be discontinuous at
these points.
In constructing a graph of a continuous function the pencil need never leave the paper, while for a
discontinuous function this is not true since there is generally a jump taking place. This is of course
merely a characteristic property and not a definition of continuity or discontinuity.
Alternative to the above definition of continuity, we can define f as continuous at x ¼ x 0 if for any
> 0we can find > 0 such that j f ðxÞ f ðx 0 Þj < whenever jx x 0 j < .Note that this is simply the
definition of limit with l ¼ f ðx 0 Þ and removal of the restriction that x 6¼ x 0 .

RIGHT- AND LEFT-HAND CONTINUITY
If f is defined only for x A x 0 , the above definition does not apply. In such case we call f continuous
(on the right)at x ¼ x 0 if lim f ðxÞ¼ f ðx 0 Þ, i.e., if f ðx 0 þÞ ¼ f ðx 0 Þ. Similarly, f is continuous (on the left)
x!x 0 þ
at x ¼ x 0 if lim f ðxÞ¼ f ðx 0 Þ, i.e., f ðx 0 Þ ¼ f ðx 0 Þ. Definitions in terms of and can be given.
x!x 0

CONTINUITY IN AN INTERVAL
A function f is said to be continuous in an interval if it is continuous at all points of the interval. In
particular, if f is defined in the closed interval a @ x @ b or ½a; bŠ, then f is continuous in the interval if
and only if lim f ðxÞ¼ f ðx 0 Þ for a < x 0 < b, lim f ðxÞ¼ f ðaÞ and lim f ðxÞ¼ f ðbÞ.
x!x 0 x!aþ
x!b

THEOREMS ON CONTINUITY
Theorem 1. If f and g are continuous at x ¼ x 0 ,so also are the functions whose image values satisfy the
relations f ðxÞþ gðxÞ, f ðxÞ gðxÞ, f ðxÞgðxÞ and f ðxÞ , the last only if gðx 0 Þ 6¼ 0. Similar results hold for
gðxÞ
continuity in an interval.

Theorem 2. Functions described as follows are continuous in every finite interval: (a) all polynomials;
x
(b) sin x and cos x; (c) a ; a > 0
Theorem 3. Let the function f be continuous at the domain value x ¼ x 0 . Also suppose that a function
g, represented by z ¼ gðyÞ,is continuous at y 0 , where y ¼ f ðxÞ (i.e., the range value of f corresponding to
x 0 is a domain value of g). Then a new function, called a composite function, f ðgÞ, represented by
z ¼ g½ f ðxފ, may be created which is continuous at its domain point x ¼ x 0 . [One says that a continuous
function of a continuous function is continuous.]

Theorem 4. If f ðxÞ is continuous in a closed interval, it is bounded in the interval.

Theorem 5. If f ðxÞ is continuous at x ¼ x 0 and f ðx 0 Þ > 0 [or f ðx 0 Þ < 0], there exists an interval about
x ¼ x 0 in which f ðxÞ > 0 [or f ðxÞ < 0].

Theorem 6. If a function f ðxÞ is continuous in an interval and either strictly increasing or strictly
1
decreasing, the inverse function f ðxÞ is single-valued, continuous, and either strictly increasing or
strictly decreasing.